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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight .
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what do `` x '' and `` y '' represent ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers .
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is it just me , or does anyone else think that slope-intercept form is more standard that standard form ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ?
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how do you figure out what c is ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here .
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can we have a negative exponent ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two .
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why is 8 becoming a negative ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . ''
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why is n't 4.5 a negative ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you .
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are there any more forms to write an equation in ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers .
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to make a standard form equation , can i make a slope intercept form and move the mx to the y 's side ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers .
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is n't it easy to figure out the slope in standard form by using the coefficients of the y and x terms ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ?
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what does the c stand for ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers .
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is it possible in the standard form to say m=b/a ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ?
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is the slope of a line in standard form ( ax + by = c ) always equal to -a/b ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ?
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what does c stand for ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers .
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how do you translate a graph into standard form ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight .
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do you use the x and y intercepts ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form .
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how do you write 3x-2=-16 in slope intercept form ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 .
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why does sal use the equation 9x+16y=72 ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ?
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is there a difference between integers and real numbers ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out .
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how would you be able to solve the linear equation y= -3 when you do n't have an x ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations .
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why and when does the slope exchange the demonitor with the numerator ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 .
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is 9x+16y=72 , the same as writing it like 9x+16y-72=0 ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations .
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why is m used to represent the slope ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers .
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how would you simplify an equation in standard form where everything is a fraction ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers .
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could someone explain how to graph a standard form linear equation without changing it ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers .
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how do you write an equation in standard form with only 2 points ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ?
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what 's the difference between real number and integers ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a .
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how do you derive point-slope form ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations .
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if we already know what the slope is and what the y intercept is , then why not just plug the slope and y intercept into the slope intercept form and divide the y value out by 16 , rather than dividing each value out by 16 ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ?
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if we have the equation ax+by=c then -b shows the x intersect and -a shows the y intersect ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers .
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in standard form can x end up being a fraction or a negative number ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers .
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how do you write an equation in standard form with integer numbers only with the m and b being fractions ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a .
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you can go from slope point form to slope intercept but you ca n't go in reverse direction , right ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope .
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wait so the a and b values are the slope , if you put them in the format of a/b ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers .
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how do we get c in the standard form ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 .
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so to solve the problem , only x can be 0 while y is a number and vice-versa ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 .
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y=2 4x+ 3y= 14 4x+ 3 x 2 = 14 would this still work instead of making x zero as well and going through the whole process again ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers .
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in standard form ax+by=c is a should be posite or negative always ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers .
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did mathematicians have no better reason to introduce standard form ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers .
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can a , b and c in the standard form be irrational numbers ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out .
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how would you find the x and y intercepts for 4x+y=0 ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 .
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if i wanted to solve for y , would 4x be 0 , making y=0 ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers .
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would 6a + 6b = 12 be solved the same way as in standard form ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that .
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line p goes through the points [ -1,5 ] and [ 2,1 ] what is the y-intercept of line p ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ?
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why a , b , c must be integers ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers .
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what if my standard form equation is 2x - y = 17 what will my points be on my graph ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 .
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can `` x '' be negative ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ?
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i know what the x and y are , but what does the a , b , c stand for ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope .
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in what does he mean by m and b are constants ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers .
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how do you get standard form from a graph ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a .
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so what are the benefits of point-slope form ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers .
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is standard form the same as general form when converting linear equations into the slope and y-intercept ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form .
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when converting standard form to slope intercept , what do you do if something is example 2y=3x+4 ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight .
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9 why is y zero ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations .
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am i crazy or did sal make a mistake by saying that the x intercept was -8 ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers .
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what does a , b , and c , in the linear standard form , mean ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants .
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what does `` by '' mean ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope .
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what does the `` b '' represent ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers .
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would 18x-5y+0.2 be in standard form ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers .
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in standard form , should a , b and c have a gcf of 1 ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight .
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for example 10x+20y=100 should be written as x+2y=10 ?
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we 've already looked at several ways of writing linear equations . you could write it in slope-intercept form , where it would be of the form of y is equal to mx plus b , where m and b are constants . m is the coefficient on this mx term right over here and m would represent the slope . and then from b you 're able to figure out the y-intercept . the y , you 're able to figure out the y-intercept from this . literally the graph that represents the xy pairs that satisfy this equation , it would intersect the y-axis at the point x equals zero , y is equal to b . and it 's slope would be m. we 've already seen that multiple times . we 've also seen that you can also express things in point-slope form . so let me make it clear . this is slope-intercept . slope- intercept . and these are just different ways of writing the same equations . you can algebraically manipulate from one to the other . another way is point-slope . point-slope form . and in point-slope form , if you know that some , if you know that there 's an equation where the line that represents the solutions of that equation has a slope m. slope is equal to m. and if you know that x equals , x equals a , y equals b , satisfies that equation , then in point-slope form you can express the equation as y minus b is equal to m times x minus a . this is point-slope form and we do videos on that . but what i really want to get into in this video is another form . and it 's a form that you might have already seen . and that is standard form . standard . standard form . and standard form takes the shape of ax plus by is equal to c , where a , b , and c are integers . and what i want to do in this video , like we 've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at ? so let 's give a tangible example here . so let 's say i have the linear equation , it 's in standard form , 9x plus 16y is equal to 72 . and we wanted to graph this . so the thing that standard form is really good for is figuring out , not just the y-intercept , y-intercept is pretty good if you 're using slope-intercept form , but we can find out the y-intercept pretty clearly from standard form and the x-intercept . the x-intercept is n't so easy to figure out from these other forms right over here . so how do we do that ? well to figure out the x and y-intercepts , let 's just set up a little table here , x comma y , and so the x-intercept is going to happen when y is equal to zero . and the y-intercept is going to happen when x is equal to zero . so when y is zero , what is x ? so when y is zero , 16 times zero is zero , that term disappears , and you 're left with 9x is equal to 72 . so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out . this term goes away and you just have to say hey , nine times x is 72 , x would be eight . when y is equal to zero , x is eight . so the point , let 's see , y is zero , x is one , two , three , four , five , six , seven , eight . that 's this point , that right over here . this point right over here is the x-intercept . when we talk about x-intercepts we 're referring to the point where the line actually intersects the x-axis . now what about the y-intercept ? well , we said x equals zero , this disappears . and we 're left with 16y is equal to 72 . and so we could solve , we could solve that . so we could say , alright 16y is equal to 72 . and then divide both sides by 16 . we get y is equal to 72 over 16 . and let 's see , what is that equal to ? that is equal to , let 's see , they 're both divisible by eight , so that 's nine over two . or we could say it 's 4.5 . so when x is zero , y is 4.5 . and so , we could plot that point as well . x is zero , y is one , two , three , 4.5 . and just with these two points , two points are enough to graph a line , we can now graph it . so let 's do that . so let me , oops , though i was using the tool that would draw a straight line . let me see if i can ... so the line will look something like that . there you have it . i 've just graphed , i 've just graphed , this is the line that represents all the x and y pairs that satisfy the equation 9x plus 16y is equal to 72 . now , i mentioned standard form 's good at certain things and the good thing that standard form is , where it 's maybe somewhat unique relative to the other forms we looked at , is it 's very easy to figure out the x-intercept . it was very easy to figure out the x-intercept from standard form . and it was n't too hard to figure out the y-intercept either . if we looked at slope-intercept form , the y-intercept just kinda jumps out at you . at point-slope form , neither the x nor the y-intercept kind of jump out at you . the place where slope-intercept or point-slope form are frankly better is that it 's pretty easy to pick out the slope here , while in standard form you would have to do a little bit of work . you could use these two points , you could use the x and y-intercepts as two points and figure out the slope from there . so you can literally say , `` okay , if i 'm going from `` this point to this point , my change in x `` to go from eight to zero is negative eight . `` and to go from zero to 4.5 , '' i wrote that little delta there unnecessarily . let me . so when you go from eight to zero , your change in x is equal to negative eight . and to go from zero to 4.5 , your change in y is going to be 4.5 . so your slope , once you 've figured this out , you could say , `` okay , this is going to be `` change in y , 4.5 , over change in x , `` over negative 8 . '' and since i , at least i do n't like a decimal up here , let 's multiply the numerator and the denominator by two . you get negative nine over 16 . now once again , we had to do a little bit of work here . we either use these two points , it did n't just jump immediately out of this , although you might see a little bit of a pattern of what 's going on here . but you still have to think about is it negative ? is it positive ? you have to do a little bit of algebraic manipulation . or , what i typically do if i 'm looking for the slope , i actually might put this into , into one of the other forms . especially slope-intercept form . but standard form by itself , great for figuring out both the x and y-intercepts and it 's frankly not that hard to convert it to slope-intercept form . let 's do that just to make it clear . so if you start with 9x , let me do that in yellow . if we start with 9x plus 16y is equal to 72 and we want to put it in slope-intercept form , we can subtract 9x from both sides . you get 16y is equal to negative 9x , plus 72 . and then divide both sides by 16 . so divide everthing by 16 . and you 'll be left with y is equal to negative 9/16x , that 's the slope , you see it right there , plus 72 over 16 , we already figured out that 's 9/2 or 4.5 . so i could write , oh i 'll just write that as 4.5 . and this form over here , much easier to figure out the slope and , actually , the y-intercept jumps out at you . but the x-intercept is n't as obvious .
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so if nine times x is 72 , 72 divided by nine is eight . so x would be equal to eight . so once again , that was pretty easy to figure out .
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what would you use the x-intercept for ?
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if you 've ever been around young children , you 're probably aware of the close bond that exists between mother and child , and scientists refer to this bond as attachment . but what causes this attachment ? why is there such a strong bond between mother and child ? for years , scientist thought that it had to do with food . that a mom 's unique ability to feed her child is what resulted in attachment . but that seems a little cold and kind of discounts all the things that mothers provide for their children . for example , contact comfort or the comfort that a child receives from being held by their mother . in order to find out exactly what causes this bond , scientists conducted a series of studies which are the harlow monkey experiments . in these studies , baby monkeys were separated from their parents at a really young age . which is something that we might consider to be kind of controversial today . but these monkeys were then given the choice to choose between two different substitute mothers . and i should note now that even though we 're calling them mothers , we 're actually referring to two different vaguely monkey shaped structures that were placed in the cage with the baby monkey . the first alternative mother option was the wire mother . and this was a mother that had a vaguely face like shape on top of it and then it had chicken wire that was kind of wrapped up in a cylinder as the body . and in the middle of that cylinder was a feeding tube . which i 'll put here in green and so within the cage that the monkey is in , this is the mother that can provide food . the second mother in the cage was referred to as the cloth mother , and this mother was the same size and shape as the wire mother , but instead of having exposed chicken wire , it had a soft cloth blanket that was wrapped around it , and so this mother is the mother that can provide comfort . so our monkey has been placed in this cage with two mother substitutes . now , which mother do you think that the monkey is going to go to ? well , if you believe that food is the basis for attachment , then you would predict that the baby would go to the wire mother , because this is the mother with the feeding tube . this is the mother that can provide them food . on the other hand , if you think that attachment is based on things like comfort , then you would assume that the monkey would spend most of it 's time around the cloth mother , because this is the mother that has the soft blanket , this is the mother that can provide contact comfort . well it turns out that the baby monkeys overwhelmingly preferred the cloth mother . indicating that it 's comfort and not the ability to actually provide nourishment that forms the basis of attachment . in fact , these baby monkeys did n't just go to the cloth mother , they spent a large majority of their time absolutely clinging to her . in fact , when these monkeys did eventually need to eat , many of them tried to do it while clinging to the cloth mother . so they would keep part of their body wrapped around the cloth mother , while reaching over to the wire mother to try to feed from her . these monkeys simply did not want to lose contact with the cloth mother . they simply did not want to give up that comfort . and over a time these monkeys did eventually become more comfortable with their situation and they would sometimes move away from the cloth mother to explore the rest of their cage . but they would always return to the cloth mother afterwards , and because of this , we can refer to the cloth mother as being a secure base . and by that all i mean is that the baby monkey was secure in the knowledge that the cloth mother was n't going anywhere . the baby monkey knew it could leave the cloth mother to explore , but that if it became anxious , that it would still be there when they got back . and so researchers would say that this pure attachment the baby monkey had with the cloth mother allows this cloth mother to act as a secure base , which eventually makes the monkey comfortable enough to explore the world on it 's own .
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on the other hand , if you think that attachment is based on things like comfort , then you would assume that the monkey would spend most of it 's time around the cloth mother , because this is the mother that has the soft blanket , this is the mother that can provide contact comfort . well it turns out that the baby monkeys overwhelmingly preferred the cloth mother . indicating that it 's comfort and not the ability to actually provide nourishment that forms the basis of attachment .
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does the cloth 'mother ' smell to the baby monkeys ?
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if you 've ever been around young children , you 're probably aware of the close bond that exists between mother and child , and scientists refer to this bond as attachment . but what causes this attachment ? why is there such a strong bond between mother and child ? for years , scientist thought that it had to do with food . that a mom 's unique ability to feed her child is what resulted in attachment . but that seems a little cold and kind of discounts all the things that mothers provide for their children . for example , contact comfort or the comfort that a child receives from being held by their mother . in order to find out exactly what causes this bond , scientists conducted a series of studies which are the harlow monkey experiments . in these studies , baby monkeys were separated from their parents at a really young age . which is something that we might consider to be kind of controversial today . but these monkeys were then given the choice to choose between two different substitute mothers . and i should note now that even though we 're calling them mothers , we 're actually referring to two different vaguely monkey shaped structures that were placed in the cage with the baby monkey . the first alternative mother option was the wire mother . and this was a mother that had a vaguely face like shape on top of it and then it had chicken wire that was kind of wrapped up in a cylinder as the body . and in the middle of that cylinder was a feeding tube . which i 'll put here in green and so within the cage that the monkey is in , this is the mother that can provide food . the second mother in the cage was referred to as the cloth mother , and this mother was the same size and shape as the wire mother , but instead of having exposed chicken wire , it had a soft cloth blanket that was wrapped around it , and so this mother is the mother that can provide comfort . so our monkey has been placed in this cage with two mother substitutes . now , which mother do you think that the monkey is going to go to ? well , if you believe that food is the basis for attachment , then you would predict that the baby would go to the wire mother , because this is the mother with the feeding tube . this is the mother that can provide them food . on the other hand , if you think that attachment is based on things like comfort , then you would assume that the monkey would spend most of it 's time around the cloth mother , because this is the mother that has the soft blanket , this is the mother that can provide contact comfort . well it turns out that the baby monkeys overwhelmingly preferred the cloth mother . indicating that it 's comfort and not the ability to actually provide nourishment that forms the basis of attachment . in fact , these baby monkeys did n't just go to the cloth mother , they spent a large majority of their time absolutely clinging to her . in fact , when these monkeys did eventually need to eat , many of them tried to do it while clinging to the cloth mother . so they would keep part of their body wrapped around the cloth mother , while reaching over to the wire mother to try to feed from her . these monkeys simply did not want to lose contact with the cloth mother . they simply did not want to give up that comfort . and over a time these monkeys did eventually become more comfortable with their situation and they would sometimes move away from the cloth mother to explore the rest of their cage . but they would always return to the cloth mother afterwards , and because of this , we can refer to the cloth mother as being a secure base . and by that all i mean is that the baby monkey was secure in the knowledge that the cloth mother was n't going anywhere . the baby monkey knew it could leave the cloth mother to explore , but that if it became anxious , that it would still be there when they got back . and so researchers would say that this pure attachment the baby monkey had with the cloth mother allows this cloth mother to act as a secure base , which eventually makes the monkey comfortable enough to explore the world on it 's own .
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and i should note now that even though we 're calling them mothers , we 're actually referring to two different vaguely monkey shaped structures that were placed in the cage with the baby monkey . the first alternative mother option was the wire mother . and this was a mother that had a vaguely face like shape on top of it and then it had chicken wire that was kind of wrapped up in a cylinder as the body .
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was the wire 'mother ' heated or cold ?
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if you 've ever been around young children , you 're probably aware of the close bond that exists between mother and child , and scientists refer to this bond as attachment . but what causes this attachment ? why is there such a strong bond between mother and child ? for years , scientist thought that it had to do with food . that a mom 's unique ability to feed her child is what resulted in attachment . but that seems a little cold and kind of discounts all the things that mothers provide for their children . for example , contact comfort or the comfort that a child receives from being held by their mother . in order to find out exactly what causes this bond , scientists conducted a series of studies which are the harlow monkey experiments . in these studies , baby monkeys were separated from their parents at a really young age . which is something that we might consider to be kind of controversial today . but these monkeys were then given the choice to choose between two different substitute mothers . and i should note now that even though we 're calling them mothers , we 're actually referring to two different vaguely monkey shaped structures that were placed in the cage with the baby monkey . the first alternative mother option was the wire mother . and this was a mother that had a vaguely face like shape on top of it and then it had chicken wire that was kind of wrapped up in a cylinder as the body . and in the middle of that cylinder was a feeding tube . which i 'll put here in green and so within the cage that the monkey is in , this is the mother that can provide food . the second mother in the cage was referred to as the cloth mother , and this mother was the same size and shape as the wire mother , but instead of having exposed chicken wire , it had a soft cloth blanket that was wrapped around it , and so this mother is the mother that can provide comfort . so our monkey has been placed in this cage with two mother substitutes . now , which mother do you think that the monkey is going to go to ? well , if you believe that food is the basis for attachment , then you would predict that the baby would go to the wire mother , because this is the mother with the feeding tube . this is the mother that can provide them food . on the other hand , if you think that attachment is based on things like comfort , then you would assume that the monkey would spend most of it 's time around the cloth mother , because this is the mother that has the soft blanket , this is the mother that can provide contact comfort . well it turns out that the baby monkeys overwhelmingly preferred the cloth mother . indicating that it 's comfort and not the ability to actually provide nourishment that forms the basis of attachment . in fact , these baby monkeys did n't just go to the cloth mother , they spent a large majority of their time absolutely clinging to her . in fact , when these monkeys did eventually need to eat , many of them tried to do it while clinging to the cloth mother . so they would keep part of their body wrapped around the cloth mother , while reaching over to the wire mother to try to feed from her . these monkeys simply did not want to lose contact with the cloth mother . they simply did not want to give up that comfort . and over a time these monkeys did eventually become more comfortable with their situation and they would sometimes move away from the cloth mother to explore the rest of their cage . but they would always return to the cloth mother afterwards , and because of this , we can refer to the cloth mother as being a secure base . and by that all i mean is that the baby monkey was secure in the knowledge that the cloth mother was n't going anywhere . the baby monkey knew it could leave the cloth mother to explore , but that if it became anxious , that it would still be there when they got back . and so researchers would say that this pure attachment the baby monkey had with the cloth mother allows this cloth mother to act as a secure base , which eventually makes the monkey comfortable enough to explore the world on it 's own .
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for years , scientist thought that it had to do with food . that a mom 's unique ability to feed her child is what resulted in attachment . but that seems a little cold and kind of discounts all the things that mothers provide for their children .
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was there any attachment ( clinging ) whatsoever made between the infant monkey and the `` wire mom '' ?
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in this video , i 'm going to expose you to what is maybe one of at least the top five most useful formulas in mathematics . and if you 've seen many of my videos , you know that i 'm not a big fan of memorizing things . but i will recommend you memorize it with the caveat that you also remember how to prove it , because i do n't want you to just remember things and not know where they came from . but with that said , let me show you what i 'm talking about : it 's the quadratic formula . and as you might guess , it is to solve for the roots , or the zeroes of quadratic equations . so let 's speak in very general terms and i 'll show you some examples . so let 's say i have an equation of the form ax squared plus bx plus c is equal to 0 . you should recognize this . this is a quadratic equation where a , b and c are -- well , a is the coefficient on the x squared term or the second degree term , b is the coefficient on the x term and then c , is , you could imagine , the coefficient on the x to the zero term , or it 's the constant term . now , given that you have a general quadratic equation like this , the quadratic formula tells us that the solutions to this equation are x is equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . and i know it seems crazy and convoluted and hard for you to memorize right now , but as you get a lot more practice you 'll see that it actually is a pretty reasonable formula to stick in your brain someplace . and you might say , gee , this is a wacky formula , where did it come from ? and in the next video i 'm going to show you where it came from . but i want you to get used to using it first . but it really just came from completing the square on this equation right there . if you complete the square here , you 're actually going to get this solution and that is the quadratic formula , right there . so let 's apply it to some problems . let 's start off with something that we could have factored just to verify that it 's giving us the same answer . so let 's say we have x squared plus 4x minus 21 is equal to 0 . so in this situation -- let me do that in a different color -- a is equal to 1 , right ? the coefficient on the x squared term is 1. b is equal to 4 , the coefficient on the x-term . and then c is equal to negative 21 , the constant term . and let 's just plug it in the formula , so what do we get ? we get x , this tells us that x is going to be equal to negative b . negative b is negative 4 -- i put the negative sign in front of that -- negative b plus or minus the square root of b squared . b squared is 16 , right ? 4 squared is 16 , minus 4 times a , which is 1 , times c , which is negative 21 . so we can put a 21 out there and that negative sign will cancel out just like that with that -- since this is the first time we 're doing it , let me not skip too many steps . so negative 21 , just so you can see how it fit in , and then all of that over 2a . a is 1 , so all of that over 2 . so what does this simplify , or hopefully it simplifies ? so we get x is equal to negative 4 plus or minus the square root of -- let 's see we have a negative times a negative , that 's going to give us a positive . and we had 16 plus , let 's see this is 6 , 4 times 1 is 4 times 21 is 84 . 16 plus 84 is 100 . that 's nice . that 's a nice perfect square . all of that over 2 , and so this is going to be equal to negative 4 plus or minus 10 over 2 . we could just divide both of these terms by 2 right now . so this is equal to negative 4 divided by 2 is negative 2 plus or minus 10 divided by 2 is 5 . so that tells us that x could be equal to negative 2 plus 5 , which is 3 , or x could be equal to negative 2 minus 5 , which is negative 7 . so the quadratic formula seems to have given us an answer for this . you can verify just by substituting back in that these do work , or you could even just try to factor this right here . you say what two numbers when you take their product , you get negative 21 and when you take their sum you get positive 4 ? so you 'd get x plus 7 times x minus 3 is equal to negative 21 . notice 7 times negative 3 is negative 21 , 7 minus 3 is positive 4 . you would get x plus -- sorry it 's not negative -- 21 is equal to 0 . there should be a 0 there . so you get x plus 7 is equal to 0 , or x minus 3 is equal to 0 . x could be equal to negative 7 or x could be equal to 3 . so it definitely gives us the same answer as factoring , so you might say , hey why bother with this crazy mess ? and the reason we want to bother with this crazy mess is it 'll also work for problems that are hard to factor . and let 's do a couple of those , let 's do some hard-to-factor problems right now . so let 's scroll down to get some fresh real estate . let 's rewrite the formula again , just in case we have n't had it memorized yet . x is going to be equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . i 'll supply this to another problem . let 's say we have the equation 3x squared plus 6x is equal to negative 10 . well , the first thing we want to do is get it in the form where all of our terms or on the left-hand side , so let 's add 10 to both sides of this equation . we get 3x squared plus the 6x plus 10 is equal to 0 . and now we can use a quadratic formula . so let 's apply it here . so a is equal to 3 . that is a , this is b and this right here is c. so the quadratic formula tells us the solutions to this equation . the roots of this quadratic function , i guess we could call it . x is going to be equal to negative b. b is 6 , so negative 6 plus or minus the square root of b squared . b is 6 , so we get 6 squared minus 4 times a , which is 3 times c , which is 10 . let 's stretch out the radical little bit , all of that over 2 times a , 2 times 3 . so we get x is equal to negative 6 plus or minus the square root of 36 minus -- this is interesting -- minus 4 times 3 times 10 . so this is minus -- 4 times 3 times 10 . so this is minus 120 . all of that over 6 . so this is interesting , you might already realize why it 's interesting . what is this going to simplify to ? 36 minus 120 is what ? that 's 84 . we make this into a 10 , this will become an 11 , this is a 4 . it is 84 , so this is going to be equal to negative 6 plus or minus the square root of -- but not positive 84 , that 's if it 's 120 minus 36 . we have 36 minus 120 . it 's going to be negative 84 all of that 6 . so you might say , gee , this is crazy . what a this silly quadratic formula you 're introducing me to , sal ? it 's worthless . it just gives me a square root of a negative number . it 's not giving me an answer . and the reason why it 's not giving you an answer , at least an answer that you might want , is because this will have no real solutions . in the future , we 're going to introduce something called an imaginary number , which is a square root of a negative number , and then we can actually express this in terms of those numbers . so this actually does have solutions , but they involve imaginary numbers . so this actually has no real solutions , we 're taking the square root of a negative number . so the b squared with the b squared minus 4ac , if this term right here is negative , then you 're not going to have any real solutions . and let 's verify that for ourselves . let 's get our graphic calculator out and let 's graph this equation right here . so , let 's get the graphs that y is equal to -- that 's what i had there before -- 3x squared plus 6x plus 10 . so that 's the equation and we 're going to see where it intersects the x-axis . where does it equal 0 ? so let me graph it . notice , this thing just comes down and then goes back up . its vertex is sitting here above the x-axis and it 's upward-opening . it never intersects the x-axis . so at no point will this expression , will this function , equal 0 . at no point will y equal 0 on this graph . so once again , the quadratic formula seems to be working . let 's do one more example , you can never see enough examples here . and i want to do ones that are , you know , maybe not so obvious to factor . so let 's say we get negative 3x squared plus 12x plus 1 is equal to 0 . now let 's try to do it just having the quadratic formula in our brain . so the x 's that satisfy this equation are going to be negative b . this is b so negative b is negative 12 plus or minus the square root of b squared , of 144 , that 's b squared minus 4 times a , which is negative 3 times c , which is 1 , all of that over 2 times a , over 2 times negative 3 . so all of that over negative 6 , this is going to be equal to negative 12 plus or minus the square root of -- what is this ? it 's a negative times a negative so they cancel out . so i have 144 plus 12 , so that is 156 , right ? 144 plus 12 , all of that over negative 6 . now , i suspect we can simplify this 156 . we could maybe bring some things out of the radical sign . so let 's attempt to do that . so let 's do a prime factorization of 156 . sometimes , this is the hardest part , simplifying the radical . so 156 is the same thing as 2 times 78 . 78 is the same thing as 2 times what ? that 's 2 times 39 . so the square root of 156 is equal to the square root of 2 times 2 times 39 or we could say that 's the square root of 2 times 2 times the square root of 39 . and this , obviously , is just going to be the square root of 4 or this is the square root of 2 times 2 is just 2 . 2 square roots of 39 , if i did that properly , let 's see , 4 times 39 . yeah , it looks like it 's right . so this up here will simplify to negative 12 plus or minus 2 times the square root of 39 , all of that over negative 6 . now we can divide the numerator and the denominator maybe by 2 . so this will be equal to negative 6 plus or minus the square root of 39 over negative 3 . or we could separate these two terms out . we could say this is equal to negative 6 over negative 3 plus or minus the square root of 39 over negative 3 . now , this is just a 2 right here , right ? these cancel out , 6 divided by 3 is 2 , so we get 2 . and now notice , if this is plus and we use this minus sign , the plus will become negative and the negative will become positive . but it still does n't matter , right ? we could say minus or plus , that 's the same thing as plus or minus the square root of 39 nine over 3 . i think that 's about as simple as we can get this answered . i want to make a very clear point of what i did that last step . i did not forget about this negative sign . i just said it does n't matter . it 's going to turn the positive into the negative ; it 's going to turn the negative into the positive . let me rewrite this . so this right here can be rewritten as 2 plus the square root of 39 over negative 3 or 2 minus the square root of 39 over negative 3 , right ? that 's what the plus or minus means , it could be this or that or both of them , really . now in this situation , this negative 3 will turn into 2 minus the square root of 39 over 3 , right ? i 'm just taking this negative out . here the negative and the negative will become a positive , and you get 2 plus the square root of 39 over 3 , right ? a negative times a negative is a positive . so once again , you have 2 plus or minus the square of 39 over 3 . 2 plus or minus the square root of 39 over 3 are solutions to this equation right there . let verify . i 'm just curious what the graph looks like . so let 's just look at it . let me clear this . where is the clear button ? so we have negative 3 three squared plus 12x plus 1 and let 's graph it . let 's see where it intersects the x-axis . it goes up there and then back down again . so 2 plus or minus the square , you see -- the square root of 39 is going to be a little bit more than 6 , right ? because 36 is 6 squared . so it 's going be a little bit more than 6 , so this is going to be a little bit more than 2 . a little bit more than 6 divided by 2 is a little bit more than 2 . so you 're going to get one value that 's a little bit more than 4 and then another value that should be a little bit less than 1 . and that looks like the case , you have 1 , 2 , 3 , 4 . you have a value that 's pretty close to 4 , and then you have another value that is a little bit -- it looks close to 0 but maybe a little bit less than that . so anyway , hopefully you found this application of the quadratic formula helpful .
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we get 3x squared plus the 6x plus 10 is equal to 0 . and now we can use a quadratic formula . so let 's apply it here .
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what is a real-life situation where someone would need to know the quadratic formula ?
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in this video , i 'm going to expose you to what is maybe one of at least the top five most useful formulas in mathematics . and if you 've seen many of my videos , you know that i 'm not a big fan of memorizing things . but i will recommend you memorize it with the caveat that you also remember how to prove it , because i do n't want you to just remember things and not know where they came from . but with that said , let me show you what i 'm talking about : it 's the quadratic formula . and as you might guess , it is to solve for the roots , or the zeroes of quadratic equations . so let 's speak in very general terms and i 'll show you some examples . so let 's say i have an equation of the form ax squared plus bx plus c is equal to 0 . you should recognize this . this is a quadratic equation where a , b and c are -- well , a is the coefficient on the x squared term or the second degree term , b is the coefficient on the x term and then c , is , you could imagine , the coefficient on the x to the zero term , or it 's the constant term . now , given that you have a general quadratic equation like this , the quadratic formula tells us that the solutions to this equation are x is equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . and i know it seems crazy and convoluted and hard for you to memorize right now , but as you get a lot more practice you 'll see that it actually is a pretty reasonable formula to stick in your brain someplace . and you might say , gee , this is a wacky formula , where did it come from ? and in the next video i 'm going to show you where it came from . but i want you to get used to using it first . but it really just came from completing the square on this equation right there . if you complete the square here , you 're actually going to get this solution and that is the quadratic formula , right there . so let 's apply it to some problems . let 's start off with something that we could have factored just to verify that it 's giving us the same answer . so let 's say we have x squared plus 4x minus 21 is equal to 0 . so in this situation -- let me do that in a different color -- a is equal to 1 , right ? the coefficient on the x squared term is 1. b is equal to 4 , the coefficient on the x-term . and then c is equal to negative 21 , the constant term . and let 's just plug it in the formula , so what do we get ? we get x , this tells us that x is going to be equal to negative b . negative b is negative 4 -- i put the negative sign in front of that -- negative b plus or minus the square root of b squared . b squared is 16 , right ? 4 squared is 16 , minus 4 times a , which is 1 , times c , which is negative 21 . so we can put a 21 out there and that negative sign will cancel out just like that with that -- since this is the first time we 're doing it , let me not skip too many steps . so negative 21 , just so you can see how it fit in , and then all of that over 2a . a is 1 , so all of that over 2 . so what does this simplify , or hopefully it simplifies ? so we get x is equal to negative 4 plus or minus the square root of -- let 's see we have a negative times a negative , that 's going to give us a positive . and we had 16 plus , let 's see this is 6 , 4 times 1 is 4 times 21 is 84 . 16 plus 84 is 100 . that 's nice . that 's a nice perfect square . all of that over 2 , and so this is going to be equal to negative 4 plus or minus 10 over 2 . we could just divide both of these terms by 2 right now . so this is equal to negative 4 divided by 2 is negative 2 plus or minus 10 divided by 2 is 5 . so that tells us that x could be equal to negative 2 plus 5 , which is 3 , or x could be equal to negative 2 minus 5 , which is negative 7 . so the quadratic formula seems to have given us an answer for this . you can verify just by substituting back in that these do work , or you could even just try to factor this right here . you say what two numbers when you take their product , you get negative 21 and when you take their sum you get positive 4 ? so you 'd get x plus 7 times x minus 3 is equal to negative 21 . notice 7 times negative 3 is negative 21 , 7 minus 3 is positive 4 . you would get x plus -- sorry it 's not negative -- 21 is equal to 0 . there should be a 0 there . so you get x plus 7 is equal to 0 , or x minus 3 is equal to 0 . x could be equal to negative 7 or x could be equal to 3 . so it definitely gives us the same answer as factoring , so you might say , hey why bother with this crazy mess ? and the reason we want to bother with this crazy mess is it 'll also work for problems that are hard to factor . and let 's do a couple of those , let 's do some hard-to-factor problems right now . so let 's scroll down to get some fresh real estate . let 's rewrite the formula again , just in case we have n't had it memorized yet . x is going to be equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . i 'll supply this to another problem . let 's say we have the equation 3x squared plus 6x is equal to negative 10 . well , the first thing we want to do is get it in the form where all of our terms or on the left-hand side , so let 's add 10 to both sides of this equation . we get 3x squared plus the 6x plus 10 is equal to 0 . and now we can use a quadratic formula . so let 's apply it here . so a is equal to 3 . that is a , this is b and this right here is c. so the quadratic formula tells us the solutions to this equation . the roots of this quadratic function , i guess we could call it . x is going to be equal to negative b. b is 6 , so negative 6 plus or minus the square root of b squared . b is 6 , so we get 6 squared minus 4 times a , which is 3 times c , which is 10 . let 's stretch out the radical little bit , all of that over 2 times a , 2 times 3 . so we get x is equal to negative 6 plus or minus the square root of 36 minus -- this is interesting -- minus 4 times 3 times 10 . so this is minus -- 4 times 3 times 10 . so this is minus 120 . all of that over 6 . so this is interesting , you might already realize why it 's interesting . what is this going to simplify to ? 36 minus 120 is what ? that 's 84 . we make this into a 10 , this will become an 11 , this is a 4 . it is 84 , so this is going to be equal to negative 6 plus or minus the square root of -- but not positive 84 , that 's if it 's 120 minus 36 . we have 36 minus 120 . it 's going to be negative 84 all of that 6 . so you might say , gee , this is crazy . what a this silly quadratic formula you 're introducing me to , sal ? it 's worthless . it just gives me a square root of a negative number . it 's not giving me an answer . and the reason why it 's not giving you an answer , at least an answer that you might want , is because this will have no real solutions . in the future , we 're going to introduce something called an imaginary number , which is a square root of a negative number , and then we can actually express this in terms of those numbers . so this actually does have solutions , but they involve imaginary numbers . so this actually has no real solutions , we 're taking the square root of a negative number . so the b squared with the b squared minus 4ac , if this term right here is negative , then you 're not going to have any real solutions . and let 's verify that for ourselves . let 's get our graphic calculator out and let 's graph this equation right here . so , let 's get the graphs that y is equal to -- that 's what i had there before -- 3x squared plus 6x plus 10 . so that 's the equation and we 're going to see where it intersects the x-axis . where does it equal 0 ? so let me graph it . notice , this thing just comes down and then goes back up . its vertex is sitting here above the x-axis and it 's upward-opening . it never intersects the x-axis . so at no point will this expression , will this function , equal 0 . at no point will y equal 0 on this graph . so once again , the quadratic formula seems to be working . let 's do one more example , you can never see enough examples here . and i want to do ones that are , you know , maybe not so obvious to factor . so let 's say we get negative 3x squared plus 12x plus 1 is equal to 0 . now let 's try to do it just having the quadratic formula in our brain . so the x 's that satisfy this equation are going to be negative b . this is b so negative b is negative 12 plus or minus the square root of b squared , of 144 , that 's b squared minus 4 times a , which is negative 3 times c , which is 1 , all of that over 2 times a , over 2 times negative 3 . so all of that over negative 6 , this is going to be equal to negative 12 plus or minus the square root of -- what is this ? it 's a negative times a negative so they cancel out . so i have 144 plus 12 , so that is 156 , right ? 144 plus 12 , all of that over negative 6 . now , i suspect we can simplify this 156 . we could maybe bring some things out of the radical sign . so let 's attempt to do that . so let 's do a prime factorization of 156 . sometimes , this is the hardest part , simplifying the radical . so 156 is the same thing as 2 times 78 . 78 is the same thing as 2 times what ? that 's 2 times 39 . so the square root of 156 is equal to the square root of 2 times 2 times 39 or we could say that 's the square root of 2 times 2 times the square root of 39 . and this , obviously , is just going to be the square root of 4 or this is the square root of 2 times 2 is just 2 . 2 square roots of 39 , if i did that properly , let 's see , 4 times 39 . yeah , it looks like it 's right . so this up here will simplify to negative 12 plus or minus 2 times the square root of 39 , all of that over negative 6 . now we can divide the numerator and the denominator maybe by 2 . so this will be equal to negative 6 plus or minus the square root of 39 over negative 3 . or we could separate these two terms out . we could say this is equal to negative 6 over negative 3 plus or minus the square root of 39 over negative 3 . now , this is just a 2 right here , right ? these cancel out , 6 divided by 3 is 2 , so we get 2 . and now notice , if this is plus and we use this minus sign , the plus will become negative and the negative will become positive . but it still does n't matter , right ? we could say minus or plus , that 's the same thing as plus or minus the square root of 39 nine over 3 . i think that 's about as simple as we can get this answered . i want to make a very clear point of what i did that last step . i did not forget about this negative sign . i just said it does n't matter . it 's going to turn the positive into the negative ; it 's going to turn the negative into the positive . let me rewrite this . so this right here can be rewritten as 2 plus the square root of 39 over negative 3 or 2 minus the square root of 39 over negative 3 , right ? that 's what the plus or minus means , it could be this or that or both of them , really . now in this situation , this negative 3 will turn into 2 minus the square root of 39 over 3 , right ? i 'm just taking this negative out . here the negative and the negative will become a positive , and you get 2 plus the square root of 39 over 3 , right ? a negative times a negative is a positive . so once again , you have 2 plus or minus the square of 39 over 3 . 2 plus or minus the square root of 39 over 3 are solutions to this equation right there . let verify . i 'm just curious what the graph looks like . so let 's just look at it . let me clear this . where is the clear button ? so we have negative 3 three squared plus 12x plus 1 and let 's graph it . let 's see where it intersects the x-axis . it goes up there and then back down again . so 2 plus or minus the square , you see -- the square root of 39 is going to be a little bit more than 6 , right ? because 36 is 6 squared . so it 's going be a little bit more than 6 , so this is going to be a little bit more than 2 . a little bit more than 6 divided by 2 is a little bit more than 2 . so you 're going to get one value that 's a little bit more than 4 and then another value that should be a little bit less than 1 . and that looks like the case , you have 1 , 2 , 3 , 4 . you have a value that 's pretty close to 4 , and then you have another value that is a little bit -- it looks close to 0 but maybe a little bit less than that . so anyway , hopefully you found this application of the quadratic formula helpful .
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that 's 2 times 39 . so the square root of 156 is equal to the square root of 2 times 2 times 39 or we could say that 's the square root of 2 times 2 times the square root of 39 . and this , obviously , is just going to be the square root of 4 or this is the square root of 2 times 2 is just 2 .
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or is it impossible to reduce something under a square root sign ?
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in this video , i 'm going to expose you to what is maybe one of at least the top five most useful formulas in mathematics . and if you 've seen many of my videos , you know that i 'm not a big fan of memorizing things . but i will recommend you memorize it with the caveat that you also remember how to prove it , because i do n't want you to just remember things and not know where they came from . but with that said , let me show you what i 'm talking about : it 's the quadratic formula . and as you might guess , it is to solve for the roots , or the zeroes of quadratic equations . so let 's speak in very general terms and i 'll show you some examples . so let 's say i have an equation of the form ax squared plus bx plus c is equal to 0 . you should recognize this . this is a quadratic equation where a , b and c are -- well , a is the coefficient on the x squared term or the second degree term , b is the coefficient on the x term and then c , is , you could imagine , the coefficient on the x to the zero term , or it 's the constant term . now , given that you have a general quadratic equation like this , the quadratic formula tells us that the solutions to this equation are x is equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . and i know it seems crazy and convoluted and hard for you to memorize right now , but as you get a lot more practice you 'll see that it actually is a pretty reasonable formula to stick in your brain someplace . and you might say , gee , this is a wacky formula , where did it come from ? and in the next video i 'm going to show you where it came from . but i want you to get used to using it first . but it really just came from completing the square on this equation right there . if you complete the square here , you 're actually going to get this solution and that is the quadratic formula , right there . so let 's apply it to some problems . let 's start off with something that we could have factored just to verify that it 's giving us the same answer . so let 's say we have x squared plus 4x minus 21 is equal to 0 . so in this situation -- let me do that in a different color -- a is equal to 1 , right ? the coefficient on the x squared term is 1. b is equal to 4 , the coefficient on the x-term . and then c is equal to negative 21 , the constant term . and let 's just plug it in the formula , so what do we get ? we get x , this tells us that x is going to be equal to negative b . negative b is negative 4 -- i put the negative sign in front of that -- negative b plus or minus the square root of b squared . b squared is 16 , right ? 4 squared is 16 , minus 4 times a , which is 1 , times c , which is negative 21 . so we can put a 21 out there and that negative sign will cancel out just like that with that -- since this is the first time we 're doing it , let me not skip too many steps . so negative 21 , just so you can see how it fit in , and then all of that over 2a . a is 1 , so all of that over 2 . so what does this simplify , or hopefully it simplifies ? so we get x is equal to negative 4 plus or minus the square root of -- let 's see we have a negative times a negative , that 's going to give us a positive . and we had 16 plus , let 's see this is 6 , 4 times 1 is 4 times 21 is 84 . 16 plus 84 is 100 . that 's nice . that 's a nice perfect square . all of that over 2 , and so this is going to be equal to negative 4 plus or minus 10 over 2 . we could just divide both of these terms by 2 right now . so this is equal to negative 4 divided by 2 is negative 2 plus or minus 10 divided by 2 is 5 . so that tells us that x could be equal to negative 2 plus 5 , which is 3 , or x could be equal to negative 2 minus 5 , which is negative 7 . so the quadratic formula seems to have given us an answer for this . you can verify just by substituting back in that these do work , or you could even just try to factor this right here . you say what two numbers when you take their product , you get negative 21 and when you take their sum you get positive 4 ? so you 'd get x plus 7 times x minus 3 is equal to negative 21 . notice 7 times negative 3 is negative 21 , 7 minus 3 is positive 4 . you would get x plus -- sorry it 's not negative -- 21 is equal to 0 . there should be a 0 there . so you get x plus 7 is equal to 0 , or x minus 3 is equal to 0 . x could be equal to negative 7 or x could be equal to 3 . so it definitely gives us the same answer as factoring , so you might say , hey why bother with this crazy mess ? and the reason we want to bother with this crazy mess is it 'll also work for problems that are hard to factor . and let 's do a couple of those , let 's do some hard-to-factor problems right now . so let 's scroll down to get some fresh real estate . let 's rewrite the formula again , just in case we have n't had it memorized yet . x is going to be equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . i 'll supply this to another problem . let 's say we have the equation 3x squared plus 6x is equal to negative 10 . well , the first thing we want to do is get it in the form where all of our terms or on the left-hand side , so let 's add 10 to both sides of this equation . we get 3x squared plus the 6x plus 10 is equal to 0 . and now we can use a quadratic formula . so let 's apply it here . so a is equal to 3 . that is a , this is b and this right here is c. so the quadratic formula tells us the solutions to this equation . the roots of this quadratic function , i guess we could call it . x is going to be equal to negative b. b is 6 , so negative 6 plus or minus the square root of b squared . b is 6 , so we get 6 squared minus 4 times a , which is 3 times c , which is 10 . let 's stretch out the radical little bit , all of that over 2 times a , 2 times 3 . so we get x is equal to negative 6 plus or minus the square root of 36 minus -- this is interesting -- minus 4 times 3 times 10 . so this is minus -- 4 times 3 times 10 . so this is minus 120 . all of that over 6 . so this is interesting , you might already realize why it 's interesting . what is this going to simplify to ? 36 minus 120 is what ? that 's 84 . we make this into a 10 , this will become an 11 , this is a 4 . it is 84 , so this is going to be equal to negative 6 plus or minus the square root of -- but not positive 84 , that 's if it 's 120 minus 36 . we have 36 minus 120 . it 's going to be negative 84 all of that 6 . so you might say , gee , this is crazy . what a this silly quadratic formula you 're introducing me to , sal ? it 's worthless . it just gives me a square root of a negative number . it 's not giving me an answer . and the reason why it 's not giving you an answer , at least an answer that you might want , is because this will have no real solutions . in the future , we 're going to introduce something called an imaginary number , which is a square root of a negative number , and then we can actually express this in terms of those numbers . so this actually does have solutions , but they involve imaginary numbers . so this actually has no real solutions , we 're taking the square root of a negative number . so the b squared with the b squared minus 4ac , if this term right here is negative , then you 're not going to have any real solutions . and let 's verify that for ourselves . let 's get our graphic calculator out and let 's graph this equation right here . so , let 's get the graphs that y is equal to -- that 's what i had there before -- 3x squared plus 6x plus 10 . so that 's the equation and we 're going to see where it intersects the x-axis . where does it equal 0 ? so let me graph it . notice , this thing just comes down and then goes back up . its vertex is sitting here above the x-axis and it 's upward-opening . it never intersects the x-axis . so at no point will this expression , will this function , equal 0 . at no point will y equal 0 on this graph . so once again , the quadratic formula seems to be working . let 's do one more example , you can never see enough examples here . and i want to do ones that are , you know , maybe not so obvious to factor . so let 's say we get negative 3x squared plus 12x plus 1 is equal to 0 . now let 's try to do it just having the quadratic formula in our brain . so the x 's that satisfy this equation are going to be negative b . this is b so negative b is negative 12 plus or minus the square root of b squared , of 144 , that 's b squared minus 4 times a , which is negative 3 times c , which is 1 , all of that over 2 times a , over 2 times negative 3 . so all of that over negative 6 , this is going to be equal to negative 12 plus or minus the square root of -- what is this ? it 's a negative times a negative so they cancel out . so i have 144 plus 12 , so that is 156 , right ? 144 plus 12 , all of that over negative 6 . now , i suspect we can simplify this 156 . we could maybe bring some things out of the radical sign . so let 's attempt to do that . so let 's do a prime factorization of 156 . sometimes , this is the hardest part , simplifying the radical . so 156 is the same thing as 2 times 78 . 78 is the same thing as 2 times what ? that 's 2 times 39 . so the square root of 156 is equal to the square root of 2 times 2 times 39 or we could say that 's the square root of 2 times 2 times the square root of 39 . and this , obviously , is just going to be the square root of 4 or this is the square root of 2 times 2 is just 2 . 2 square roots of 39 , if i did that properly , let 's see , 4 times 39 . yeah , it looks like it 's right . so this up here will simplify to negative 12 plus or minus 2 times the square root of 39 , all of that over negative 6 . now we can divide the numerator and the denominator maybe by 2 . so this will be equal to negative 6 plus or minus the square root of 39 over negative 3 . or we could separate these two terms out . we could say this is equal to negative 6 over negative 3 plus or minus the square root of 39 over negative 3 . now , this is just a 2 right here , right ? these cancel out , 6 divided by 3 is 2 , so we get 2 . and now notice , if this is plus and we use this minus sign , the plus will become negative and the negative will become positive . but it still does n't matter , right ? we could say minus or plus , that 's the same thing as plus or minus the square root of 39 nine over 3 . i think that 's about as simple as we can get this answered . i want to make a very clear point of what i did that last step . i did not forget about this negative sign . i just said it does n't matter . it 's going to turn the positive into the negative ; it 's going to turn the negative into the positive . let me rewrite this . so this right here can be rewritten as 2 plus the square root of 39 over negative 3 or 2 minus the square root of 39 over negative 3 , right ? that 's what the plus or minus means , it could be this or that or both of them , really . now in this situation , this negative 3 will turn into 2 minus the square root of 39 over 3 , right ? i 'm just taking this negative out . here the negative and the negative will become a positive , and you get 2 plus the square root of 39 over 3 , right ? a negative times a negative is a positive . so once again , you have 2 plus or minus the square of 39 over 3 . 2 plus or minus the square root of 39 over 3 are solutions to this equation right there . let verify . i 'm just curious what the graph looks like . so let 's just look at it . let me clear this . where is the clear button ? so we have negative 3 three squared plus 12x plus 1 and let 's graph it . let 's see where it intersects the x-axis . it goes up there and then back down again . so 2 plus or minus the square , you see -- the square root of 39 is going to be a little bit more than 6 , right ? because 36 is 6 squared . so it 's going be a little bit more than 6 , so this is going to be a little bit more than 2 . a little bit more than 6 divided by 2 is a little bit more than 2 . so you 're going to get one value that 's a little bit more than 4 and then another value that should be a little bit less than 1 . and that looks like the case , you have 1 , 2 , 3 , 4 . you have a value that 's pretty close to 4 , and then you have another value that is a little bit -- it looks close to 0 but maybe a little bit less than that . so anyway , hopefully you found this application of the quadratic formula helpful .
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we get 3x squared plus the 6x plus 10 is equal to 0 . and now we can use a quadratic formula . so let 's apply it here .
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when graphing quadratic equations and also using the formula we look for x , would there be a formula for finding the y intercepts ?
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in this video , i 'm going to expose you to what is maybe one of at least the top five most useful formulas in mathematics . and if you 've seen many of my videos , you know that i 'm not a big fan of memorizing things . but i will recommend you memorize it with the caveat that you also remember how to prove it , because i do n't want you to just remember things and not know where they came from . but with that said , let me show you what i 'm talking about : it 's the quadratic formula . and as you might guess , it is to solve for the roots , or the zeroes of quadratic equations . so let 's speak in very general terms and i 'll show you some examples . so let 's say i have an equation of the form ax squared plus bx plus c is equal to 0 . you should recognize this . this is a quadratic equation where a , b and c are -- well , a is the coefficient on the x squared term or the second degree term , b is the coefficient on the x term and then c , is , you could imagine , the coefficient on the x to the zero term , or it 's the constant term . now , given that you have a general quadratic equation like this , the quadratic formula tells us that the solutions to this equation are x is equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . and i know it seems crazy and convoluted and hard for you to memorize right now , but as you get a lot more practice you 'll see that it actually is a pretty reasonable formula to stick in your brain someplace . and you might say , gee , this is a wacky formula , where did it come from ? and in the next video i 'm going to show you where it came from . but i want you to get used to using it first . but it really just came from completing the square on this equation right there . if you complete the square here , you 're actually going to get this solution and that is the quadratic formula , right there . so let 's apply it to some problems . let 's start off with something that we could have factored just to verify that it 's giving us the same answer . so let 's say we have x squared plus 4x minus 21 is equal to 0 . so in this situation -- let me do that in a different color -- a is equal to 1 , right ? the coefficient on the x squared term is 1. b is equal to 4 , the coefficient on the x-term . and then c is equal to negative 21 , the constant term . and let 's just plug it in the formula , so what do we get ? we get x , this tells us that x is going to be equal to negative b . negative b is negative 4 -- i put the negative sign in front of that -- negative b plus or minus the square root of b squared . b squared is 16 , right ? 4 squared is 16 , minus 4 times a , which is 1 , times c , which is negative 21 . so we can put a 21 out there and that negative sign will cancel out just like that with that -- since this is the first time we 're doing it , let me not skip too many steps . so negative 21 , just so you can see how it fit in , and then all of that over 2a . a is 1 , so all of that over 2 . so what does this simplify , or hopefully it simplifies ? so we get x is equal to negative 4 plus or minus the square root of -- let 's see we have a negative times a negative , that 's going to give us a positive . and we had 16 plus , let 's see this is 6 , 4 times 1 is 4 times 21 is 84 . 16 plus 84 is 100 . that 's nice . that 's a nice perfect square . all of that over 2 , and so this is going to be equal to negative 4 plus or minus 10 over 2 . we could just divide both of these terms by 2 right now . so this is equal to negative 4 divided by 2 is negative 2 plus or minus 10 divided by 2 is 5 . so that tells us that x could be equal to negative 2 plus 5 , which is 3 , or x could be equal to negative 2 minus 5 , which is negative 7 . so the quadratic formula seems to have given us an answer for this . you can verify just by substituting back in that these do work , or you could even just try to factor this right here . you say what two numbers when you take their product , you get negative 21 and when you take their sum you get positive 4 ? so you 'd get x plus 7 times x minus 3 is equal to negative 21 . notice 7 times negative 3 is negative 21 , 7 minus 3 is positive 4 . you would get x plus -- sorry it 's not negative -- 21 is equal to 0 . there should be a 0 there . so you get x plus 7 is equal to 0 , or x minus 3 is equal to 0 . x could be equal to negative 7 or x could be equal to 3 . so it definitely gives us the same answer as factoring , so you might say , hey why bother with this crazy mess ? and the reason we want to bother with this crazy mess is it 'll also work for problems that are hard to factor . and let 's do a couple of those , let 's do some hard-to-factor problems right now . so let 's scroll down to get some fresh real estate . let 's rewrite the formula again , just in case we have n't had it memorized yet . x is going to be equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . i 'll supply this to another problem . let 's say we have the equation 3x squared plus 6x is equal to negative 10 . well , the first thing we want to do is get it in the form where all of our terms or on the left-hand side , so let 's add 10 to both sides of this equation . we get 3x squared plus the 6x plus 10 is equal to 0 . and now we can use a quadratic formula . so let 's apply it here . so a is equal to 3 . that is a , this is b and this right here is c. so the quadratic formula tells us the solutions to this equation . the roots of this quadratic function , i guess we could call it . x is going to be equal to negative b. b is 6 , so negative 6 plus or minus the square root of b squared . b is 6 , so we get 6 squared minus 4 times a , which is 3 times c , which is 10 . let 's stretch out the radical little bit , all of that over 2 times a , 2 times 3 . so we get x is equal to negative 6 plus or minus the square root of 36 minus -- this is interesting -- minus 4 times 3 times 10 . so this is minus -- 4 times 3 times 10 . so this is minus 120 . all of that over 6 . so this is interesting , you might already realize why it 's interesting . what is this going to simplify to ? 36 minus 120 is what ? that 's 84 . we make this into a 10 , this will become an 11 , this is a 4 . it is 84 , so this is going to be equal to negative 6 plus or minus the square root of -- but not positive 84 , that 's if it 's 120 minus 36 . we have 36 minus 120 . it 's going to be negative 84 all of that 6 . so you might say , gee , this is crazy . what a this silly quadratic formula you 're introducing me to , sal ? it 's worthless . it just gives me a square root of a negative number . it 's not giving me an answer . and the reason why it 's not giving you an answer , at least an answer that you might want , is because this will have no real solutions . in the future , we 're going to introduce something called an imaginary number , which is a square root of a negative number , and then we can actually express this in terms of those numbers . so this actually does have solutions , but they involve imaginary numbers . so this actually has no real solutions , we 're taking the square root of a negative number . so the b squared with the b squared minus 4ac , if this term right here is negative , then you 're not going to have any real solutions . and let 's verify that for ourselves . let 's get our graphic calculator out and let 's graph this equation right here . so , let 's get the graphs that y is equal to -- that 's what i had there before -- 3x squared plus 6x plus 10 . so that 's the equation and we 're going to see where it intersects the x-axis . where does it equal 0 ? so let me graph it . notice , this thing just comes down and then goes back up . its vertex is sitting here above the x-axis and it 's upward-opening . it never intersects the x-axis . so at no point will this expression , will this function , equal 0 . at no point will y equal 0 on this graph . so once again , the quadratic formula seems to be working . let 's do one more example , you can never see enough examples here . and i want to do ones that are , you know , maybe not so obvious to factor . so let 's say we get negative 3x squared plus 12x plus 1 is equal to 0 . now let 's try to do it just having the quadratic formula in our brain . so the x 's that satisfy this equation are going to be negative b . this is b so negative b is negative 12 plus or minus the square root of b squared , of 144 , that 's b squared minus 4 times a , which is negative 3 times c , which is 1 , all of that over 2 times a , over 2 times negative 3 . so all of that over negative 6 , this is going to be equal to negative 12 plus or minus the square root of -- what is this ? it 's a negative times a negative so they cancel out . so i have 144 plus 12 , so that is 156 , right ? 144 plus 12 , all of that over negative 6 . now , i suspect we can simplify this 156 . we could maybe bring some things out of the radical sign . so let 's attempt to do that . so let 's do a prime factorization of 156 . sometimes , this is the hardest part , simplifying the radical . so 156 is the same thing as 2 times 78 . 78 is the same thing as 2 times what ? that 's 2 times 39 . so the square root of 156 is equal to the square root of 2 times 2 times 39 or we could say that 's the square root of 2 times 2 times the square root of 39 . and this , obviously , is just going to be the square root of 4 or this is the square root of 2 times 2 is just 2 . 2 square roots of 39 , if i did that properly , let 's see , 4 times 39 . yeah , it looks like it 's right . so this up here will simplify to negative 12 plus or minus 2 times the square root of 39 , all of that over negative 6 . now we can divide the numerator and the denominator maybe by 2 . so this will be equal to negative 6 plus or minus the square root of 39 over negative 3 . or we could separate these two terms out . we could say this is equal to negative 6 over negative 3 plus or minus the square root of 39 over negative 3 . now , this is just a 2 right here , right ? these cancel out , 6 divided by 3 is 2 , so we get 2 . and now notice , if this is plus and we use this minus sign , the plus will become negative and the negative will become positive . but it still does n't matter , right ? we could say minus or plus , that 's the same thing as plus or minus the square root of 39 nine over 3 . i think that 's about as simple as we can get this answered . i want to make a very clear point of what i did that last step . i did not forget about this negative sign . i just said it does n't matter . it 's going to turn the positive into the negative ; it 's going to turn the negative into the positive . let me rewrite this . so this right here can be rewritten as 2 plus the square root of 39 over negative 3 or 2 minus the square root of 39 over negative 3 , right ? that 's what the plus or minus means , it could be this or that or both of them , really . now in this situation , this negative 3 will turn into 2 minus the square root of 39 over 3 , right ? i 'm just taking this negative out . here the negative and the negative will become a positive , and you get 2 plus the square root of 39 over 3 , right ? a negative times a negative is a positive . so once again , you have 2 plus or minus the square of 39 over 3 . 2 plus or minus the square root of 39 over 3 are solutions to this equation right there . let verify . i 'm just curious what the graph looks like . so let 's just look at it . let me clear this . where is the clear button ? so we have negative 3 three squared plus 12x plus 1 and let 's graph it . let 's see where it intersects the x-axis . it goes up there and then back down again . so 2 plus or minus the square , you see -- the square root of 39 is going to be a little bit more than 6 , right ? because 36 is 6 squared . so it 's going be a little bit more than 6 , so this is going to be a little bit more than 2 . a little bit more than 6 divided by 2 is a little bit more than 2 . so you 're going to get one value that 's a little bit more than 4 and then another value that should be a little bit less than 1 . and that looks like the case , you have 1 , 2 , 3 , 4 . you have a value that 's pretty close to 4 , and then you have another value that is a little bit -- it looks close to 0 but maybe a little bit less than that . so anyway , hopefully you found this application of the quadratic formula helpful .
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this is a quadratic equation where a , b and c are -- well , a is the coefficient on the x squared term or the second degree term , b is the coefficient on the x term and then c , is , you could imagine , the coefficient on the x to the zero term , or it 's the constant term . now , given that you have a general quadratic equation like this , the quadratic formula tells us that the solutions to this equation are x is equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . and i know it seems crazy and convoluted and hard for you to memorize right now , but as you get a lot more practice you 'll see that it actually is a pretty reasonable formula to stick in your brain someplace .
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how to find the quadratic equation when the roots are given ?
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in this video , i 'm going to expose you to what is maybe one of at least the top five most useful formulas in mathematics . and if you 've seen many of my videos , you know that i 'm not a big fan of memorizing things . but i will recommend you memorize it with the caveat that you also remember how to prove it , because i do n't want you to just remember things and not know where they came from . but with that said , let me show you what i 'm talking about : it 's the quadratic formula . and as you might guess , it is to solve for the roots , or the zeroes of quadratic equations . so let 's speak in very general terms and i 'll show you some examples . so let 's say i have an equation of the form ax squared plus bx plus c is equal to 0 . you should recognize this . this is a quadratic equation where a , b and c are -- well , a is the coefficient on the x squared term or the second degree term , b is the coefficient on the x term and then c , is , you could imagine , the coefficient on the x to the zero term , or it 's the constant term . now , given that you have a general quadratic equation like this , the quadratic formula tells us that the solutions to this equation are x is equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . and i know it seems crazy and convoluted and hard for you to memorize right now , but as you get a lot more practice you 'll see that it actually is a pretty reasonable formula to stick in your brain someplace . and you might say , gee , this is a wacky formula , where did it come from ? and in the next video i 'm going to show you where it came from . but i want you to get used to using it first . but it really just came from completing the square on this equation right there . if you complete the square here , you 're actually going to get this solution and that is the quadratic formula , right there . so let 's apply it to some problems . let 's start off with something that we could have factored just to verify that it 's giving us the same answer . so let 's say we have x squared plus 4x minus 21 is equal to 0 . so in this situation -- let me do that in a different color -- a is equal to 1 , right ? the coefficient on the x squared term is 1. b is equal to 4 , the coefficient on the x-term . and then c is equal to negative 21 , the constant term . and let 's just plug it in the formula , so what do we get ? we get x , this tells us that x is going to be equal to negative b . negative b is negative 4 -- i put the negative sign in front of that -- negative b plus or minus the square root of b squared . b squared is 16 , right ? 4 squared is 16 , minus 4 times a , which is 1 , times c , which is negative 21 . so we can put a 21 out there and that negative sign will cancel out just like that with that -- since this is the first time we 're doing it , let me not skip too many steps . so negative 21 , just so you can see how it fit in , and then all of that over 2a . a is 1 , so all of that over 2 . so what does this simplify , or hopefully it simplifies ? so we get x is equal to negative 4 plus or minus the square root of -- let 's see we have a negative times a negative , that 's going to give us a positive . and we had 16 plus , let 's see this is 6 , 4 times 1 is 4 times 21 is 84 . 16 plus 84 is 100 . that 's nice . that 's a nice perfect square . all of that over 2 , and so this is going to be equal to negative 4 plus or minus 10 over 2 . we could just divide both of these terms by 2 right now . so this is equal to negative 4 divided by 2 is negative 2 plus or minus 10 divided by 2 is 5 . so that tells us that x could be equal to negative 2 plus 5 , which is 3 , or x could be equal to negative 2 minus 5 , which is negative 7 . so the quadratic formula seems to have given us an answer for this . you can verify just by substituting back in that these do work , or you could even just try to factor this right here . you say what two numbers when you take their product , you get negative 21 and when you take their sum you get positive 4 ? so you 'd get x plus 7 times x minus 3 is equal to negative 21 . notice 7 times negative 3 is negative 21 , 7 minus 3 is positive 4 . you would get x plus -- sorry it 's not negative -- 21 is equal to 0 . there should be a 0 there . so you get x plus 7 is equal to 0 , or x minus 3 is equal to 0 . x could be equal to negative 7 or x could be equal to 3 . so it definitely gives us the same answer as factoring , so you might say , hey why bother with this crazy mess ? and the reason we want to bother with this crazy mess is it 'll also work for problems that are hard to factor . and let 's do a couple of those , let 's do some hard-to-factor problems right now . so let 's scroll down to get some fresh real estate . let 's rewrite the formula again , just in case we have n't had it memorized yet . x is going to be equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . i 'll supply this to another problem . let 's say we have the equation 3x squared plus 6x is equal to negative 10 . well , the first thing we want to do is get it in the form where all of our terms or on the left-hand side , so let 's add 10 to both sides of this equation . we get 3x squared plus the 6x plus 10 is equal to 0 . and now we can use a quadratic formula . so let 's apply it here . so a is equal to 3 . that is a , this is b and this right here is c. so the quadratic formula tells us the solutions to this equation . the roots of this quadratic function , i guess we could call it . x is going to be equal to negative b. b is 6 , so negative 6 plus or minus the square root of b squared . b is 6 , so we get 6 squared minus 4 times a , which is 3 times c , which is 10 . let 's stretch out the radical little bit , all of that over 2 times a , 2 times 3 . so we get x is equal to negative 6 plus or minus the square root of 36 minus -- this is interesting -- minus 4 times 3 times 10 . so this is minus -- 4 times 3 times 10 . so this is minus 120 . all of that over 6 . so this is interesting , you might already realize why it 's interesting . what is this going to simplify to ? 36 minus 120 is what ? that 's 84 . we make this into a 10 , this will become an 11 , this is a 4 . it is 84 , so this is going to be equal to negative 6 plus or minus the square root of -- but not positive 84 , that 's if it 's 120 minus 36 . we have 36 minus 120 . it 's going to be negative 84 all of that 6 . so you might say , gee , this is crazy . what a this silly quadratic formula you 're introducing me to , sal ? it 's worthless . it just gives me a square root of a negative number . it 's not giving me an answer . and the reason why it 's not giving you an answer , at least an answer that you might want , is because this will have no real solutions . in the future , we 're going to introduce something called an imaginary number , which is a square root of a negative number , and then we can actually express this in terms of those numbers . so this actually does have solutions , but they involve imaginary numbers . so this actually has no real solutions , we 're taking the square root of a negative number . so the b squared with the b squared minus 4ac , if this term right here is negative , then you 're not going to have any real solutions . and let 's verify that for ourselves . let 's get our graphic calculator out and let 's graph this equation right here . so , let 's get the graphs that y is equal to -- that 's what i had there before -- 3x squared plus 6x plus 10 . so that 's the equation and we 're going to see where it intersects the x-axis . where does it equal 0 ? so let me graph it . notice , this thing just comes down and then goes back up . its vertex is sitting here above the x-axis and it 's upward-opening . it never intersects the x-axis . so at no point will this expression , will this function , equal 0 . at no point will y equal 0 on this graph . so once again , the quadratic formula seems to be working . let 's do one more example , you can never see enough examples here . and i want to do ones that are , you know , maybe not so obvious to factor . so let 's say we get negative 3x squared plus 12x plus 1 is equal to 0 . now let 's try to do it just having the quadratic formula in our brain . so the x 's that satisfy this equation are going to be negative b . this is b so negative b is negative 12 plus or minus the square root of b squared , of 144 , that 's b squared minus 4 times a , which is negative 3 times c , which is 1 , all of that over 2 times a , over 2 times negative 3 . so all of that over negative 6 , this is going to be equal to negative 12 plus or minus the square root of -- what is this ? it 's a negative times a negative so they cancel out . so i have 144 plus 12 , so that is 156 , right ? 144 plus 12 , all of that over negative 6 . now , i suspect we can simplify this 156 . we could maybe bring some things out of the radical sign . so let 's attempt to do that . so let 's do a prime factorization of 156 . sometimes , this is the hardest part , simplifying the radical . so 156 is the same thing as 2 times 78 . 78 is the same thing as 2 times what ? that 's 2 times 39 . so the square root of 156 is equal to the square root of 2 times 2 times 39 or we could say that 's the square root of 2 times 2 times the square root of 39 . and this , obviously , is just going to be the square root of 4 or this is the square root of 2 times 2 is just 2 . 2 square roots of 39 , if i did that properly , let 's see , 4 times 39 . yeah , it looks like it 's right . so this up here will simplify to negative 12 plus or minus 2 times the square root of 39 , all of that over negative 6 . now we can divide the numerator and the denominator maybe by 2 . so this will be equal to negative 6 plus or minus the square root of 39 over negative 3 . or we could separate these two terms out . we could say this is equal to negative 6 over negative 3 plus or minus the square root of 39 over negative 3 . now , this is just a 2 right here , right ? these cancel out , 6 divided by 3 is 2 , so we get 2 . and now notice , if this is plus and we use this minus sign , the plus will become negative and the negative will become positive . but it still does n't matter , right ? we could say minus or plus , that 's the same thing as plus or minus the square root of 39 nine over 3 . i think that 's about as simple as we can get this answered . i want to make a very clear point of what i did that last step . i did not forget about this negative sign . i just said it does n't matter . it 's going to turn the positive into the negative ; it 's going to turn the negative into the positive . let me rewrite this . so this right here can be rewritten as 2 plus the square root of 39 over negative 3 or 2 minus the square root of 39 over negative 3 , right ? that 's what the plus or minus means , it could be this or that or both of them , really . now in this situation , this negative 3 will turn into 2 minus the square root of 39 over 3 , right ? i 'm just taking this negative out . here the negative and the negative will become a positive , and you get 2 plus the square root of 39 over 3 , right ? a negative times a negative is a positive . so once again , you have 2 plus or minus the square of 39 over 3 . 2 plus or minus the square root of 39 over 3 are solutions to this equation right there . let verify . i 'm just curious what the graph looks like . so let 's just look at it . let me clear this . where is the clear button ? so we have negative 3 three squared plus 12x plus 1 and let 's graph it . let 's see where it intersects the x-axis . it goes up there and then back down again . so 2 plus or minus the square , you see -- the square root of 39 is going to be a little bit more than 6 , right ? because 36 is 6 squared . so it 's going be a little bit more than 6 , so this is going to be a little bit more than 2 . a little bit more than 6 divided by 2 is a little bit more than 2 . so you 're going to get one value that 's a little bit more than 4 and then another value that should be a little bit less than 1 . and that looks like the case , you have 1 , 2 , 3 , 4 . you have a value that 's pretty close to 4 , and then you have another value that is a little bit -- it looks close to 0 but maybe a little bit less than that . so anyway , hopefully you found this application of the quadratic formula helpful .
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in the future , we 're going to introduce something called an imaginary number , which is a square root of a negative number , and then we can actually express this in terms of those numbers . so this actually does have solutions , but they involve imaginary numbers . so this actually has no real solutions , we 're taking the square root of a negative number .
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how difficult is it when you start using imaginary numbers ?
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in this video , i 'm going to expose you to what is maybe one of at least the top five most useful formulas in mathematics . and if you 've seen many of my videos , you know that i 'm not a big fan of memorizing things . but i will recommend you memorize it with the caveat that you also remember how to prove it , because i do n't want you to just remember things and not know where they came from . but with that said , let me show you what i 'm talking about : it 's the quadratic formula . and as you might guess , it is to solve for the roots , or the zeroes of quadratic equations . so let 's speak in very general terms and i 'll show you some examples . so let 's say i have an equation of the form ax squared plus bx plus c is equal to 0 . you should recognize this . this is a quadratic equation where a , b and c are -- well , a is the coefficient on the x squared term or the second degree term , b is the coefficient on the x term and then c , is , you could imagine , the coefficient on the x to the zero term , or it 's the constant term . now , given that you have a general quadratic equation like this , the quadratic formula tells us that the solutions to this equation are x is equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . and i know it seems crazy and convoluted and hard for you to memorize right now , but as you get a lot more practice you 'll see that it actually is a pretty reasonable formula to stick in your brain someplace . and you might say , gee , this is a wacky formula , where did it come from ? and in the next video i 'm going to show you where it came from . but i want you to get used to using it first . but it really just came from completing the square on this equation right there . if you complete the square here , you 're actually going to get this solution and that is the quadratic formula , right there . so let 's apply it to some problems . let 's start off with something that we could have factored just to verify that it 's giving us the same answer . so let 's say we have x squared plus 4x minus 21 is equal to 0 . so in this situation -- let me do that in a different color -- a is equal to 1 , right ? the coefficient on the x squared term is 1. b is equal to 4 , the coefficient on the x-term . and then c is equal to negative 21 , the constant term . and let 's just plug it in the formula , so what do we get ? we get x , this tells us that x is going to be equal to negative b . negative b is negative 4 -- i put the negative sign in front of that -- negative b plus or minus the square root of b squared . b squared is 16 , right ? 4 squared is 16 , minus 4 times a , which is 1 , times c , which is negative 21 . so we can put a 21 out there and that negative sign will cancel out just like that with that -- since this is the first time we 're doing it , let me not skip too many steps . so negative 21 , just so you can see how it fit in , and then all of that over 2a . a is 1 , so all of that over 2 . so what does this simplify , or hopefully it simplifies ? so we get x is equal to negative 4 plus or minus the square root of -- let 's see we have a negative times a negative , that 's going to give us a positive . and we had 16 plus , let 's see this is 6 , 4 times 1 is 4 times 21 is 84 . 16 plus 84 is 100 . that 's nice . that 's a nice perfect square . all of that over 2 , and so this is going to be equal to negative 4 plus or minus 10 over 2 . we could just divide both of these terms by 2 right now . so this is equal to negative 4 divided by 2 is negative 2 plus or minus 10 divided by 2 is 5 . so that tells us that x could be equal to negative 2 plus 5 , which is 3 , or x could be equal to negative 2 minus 5 , which is negative 7 . so the quadratic formula seems to have given us an answer for this . you can verify just by substituting back in that these do work , or you could even just try to factor this right here . you say what two numbers when you take their product , you get negative 21 and when you take their sum you get positive 4 ? so you 'd get x plus 7 times x minus 3 is equal to negative 21 . notice 7 times negative 3 is negative 21 , 7 minus 3 is positive 4 . you would get x plus -- sorry it 's not negative -- 21 is equal to 0 . there should be a 0 there . so you get x plus 7 is equal to 0 , or x minus 3 is equal to 0 . x could be equal to negative 7 or x could be equal to 3 . so it definitely gives us the same answer as factoring , so you might say , hey why bother with this crazy mess ? and the reason we want to bother with this crazy mess is it 'll also work for problems that are hard to factor . and let 's do a couple of those , let 's do some hard-to-factor problems right now . so let 's scroll down to get some fresh real estate . let 's rewrite the formula again , just in case we have n't had it memorized yet . x is going to be equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . i 'll supply this to another problem . let 's say we have the equation 3x squared plus 6x is equal to negative 10 . well , the first thing we want to do is get it in the form where all of our terms or on the left-hand side , so let 's add 10 to both sides of this equation . we get 3x squared plus the 6x plus 10 is equal to 0 . and now we can use a quadratic formula . so let 's apply it here . so a is equal to 3 . that is a , this is b and this right here is c. so the quadratic formula tells us the solutions to this equation . the roots of this quadratic function , i guess we could call it . x is going to be equal to negative b. b is 6 , so negative 6 plus or minus the square root of b squared . b is 6 , so we get 6 squared minus 4 times a , which is 3 times c , which is 10 . let 's stretch out the radical little bit , all of that over 2 times a , 2 times 3 . so we get x is equal to negative 6 plus or minus the square root of 36 minus -- this is interesting -- minus 4 times 3 times 10 . so this is minus -- 4 times 3 times 10 . so this is minus 120 . all of that over 6 . so this is interesting , you might already realize why it 's interesting . what is this going to simplify to ? 36 minus 120 is what ? that 's 84 . we make this into a 10 , this will become an 11 , this is a 4 . it is 84 , so this is going to be equal to negative 6 plus or minus the square root of -- but not positive 84 , that 's if it 's 120 minus 36 . we have 36 minus 120 . it 's going to be negative 84 all of that 6 . so you might say , gee , this is crazy . what a this silly quadratic formula you 're introducing me to , sal ? it 's worthless . it just gives me a square root of a negative number . it 's not giving me an answer . and the reason why it 's not giving you an answer , at least an answer that you might want , is because this will have no real solutions . in the future , we 're going to introduce something called an imaginary number , which is a square root of a negative number , and then we can actually express this in terms of those numbers . so this actually does have solutions , but they involve imaginary numbers . so this actually has no real solutions , we 're taking the square root of a negative number . so the b squared with the b squared minus 4ac , if this term right here is negative , then you 're not going to have any real solutions . and let 's verify that for ourselves . let 's get our graphic calculator out and let 's graph this equation right here . so , let 's get the graphs that y is equal to -- that 's what i had there before -- 3x squared plus 6x plus 10 . so that 's the equation and we 're going to see where it intersects the x-axis . where does it equal 0 ? so let me graph it . notice , this thing just comes down and then goes back up . its vertex is sitting here above the x-axis and it 's upward-opening . it never intersects the x-axis . so at no point will this expression , will this function , equal 0 . at no point will y equal 0 on this graph . so once again , the quadratic formula seems to be working . let 's do one more example , you can never see enough examples here . and i want to do ones that are , you know , maybe not so obvious to factor . so let 's say we get negative 3x squared plus 12x plus 1 is equal to 0 . now let 's try to do it just having the quadratic formula in our brain . so the x 's that satisfy this equation are going to be negative b . this is b so negative b is negative 12 plus or minus the square root of b squared , of 144 , that 's b squared minus 4 times a , which is negative 3 times c , which is 1 , all of that over 2 times a , over 2 times negative 3 . so all of that over negative 6 , this is going to be equal to negative 12 plus or minus the square root of -- what is this ? it 's a negative times a negative so they cancel out . so i have 144 plus 12 , so that is 156 , right ? 144 plus 12 , all of that over negative 6 . now , i suspect we can simplify this 156 . we could maybe bring some things out of the radical sign . so let 's attempt to do that . so let 's do a prime factorization of 156 . sometimes , this is the hardest part , simplifying the radical . so 156 is the same thing as 2 times 78 . 78 is the same thing as 2 times what ? that 's 2 times 39 . so the square root of 156 is equal to the square root of 2 times 2 times 39 or we could say that 's the square root of 2 times 2 times the square root of 39 . and this , obviously , is just going to be the square root of 4 or this is the square root of 2 times 2 is just 2 . 2 square roots of 39 , if i did that properly , let 's see , 4 times 39 . yeah , it looks like it 's right . so this up here will simplify to negative 12 plus or minus 2 times the square root of 39 , all of that over negative 6 . now we can divide the numerator and the denominator maybe by 2 . so this will be equal to negative 6 plus or minus the square root of 39 over negative 3 . or we could separate these two terms out . we could say this is equal to negative 6 over negative 3 plus or minus the square root of 39 over negative 3 . now , this is just a 2 right here , right ? these cancel out , 6 divided by 3 is 2 , so we get 2 . and now notice , if this is plus and we use this minus sign , the plus will become negative and the negative will become positive . but it still does n't matter , right ? we could say minus or plus , that 's the same thing as plus or minus the square root of 39 nine over 3 . i think that 's about as simple as we can get this answered . i want to make a very clear point of what i did that last step . i did not forget about this negative sign . i just said it does n't matter . it 's going to turn the positive into the negative ; it 's going to turn the negative into the positive . let me rewrite this . so this right here can be rewritten as 2 plus the square root of 39 over negative 3 or 2 minus the square root of 39 over negative 3 , right ? that 's what the plus or minus means , it could be this or that or both of them , really . now in this situation , this negative 3 will turn into 2 minus the square root of 39 over 3 , right ? i 'm just taking this negative out . here the negative and the negative will become a positive , and you get 2 plus the square root of 39 over 3 , right ? a negative times a negative is a positive . so once again , you have 2 plus or minus the square of 39 over 3 . 2 plus or minus the square root of 39 over 3 are solutions to this equation right there . let verify . i 'm just curious what the graph looks like . so let 's just look at it . let me clear this . where is the clear button ? so we have negative 3 three squared plus 12x plus 1 and let 's graph it . let 's see where it intersects the x-axis . it goes up there and then back down again . so 2 plus or minus the square , you see -- the square root of 39 is going to be a little bit more than 6 , right ? because 36 is 6 squared . so it 's going be a little bit more than 6 , so this is going to be a little bit more than 2 . a little bit more than 6 divided by 2 is a little bit more than 2 . so you 're going to get one value that 's a little bit more than 4 and then another value that should be a little bit less than 1 . and that looks like the case , you have 1 , 2 , 3 , 4 . you have a value that 's pretty close to 4 , and then you have another value that is a little bit -- it looks close to 0 but maybe a little bit less than that . so anyway , hopefully you found this application of the quadratic formula helpful .
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that 's 2 times 39 . so the square root of 156 is equal to the square root of 2 times 2 times 39 or we could say that 's the square root of 2 times 2 times the square root of 39 . and this , obviously , is just going to be the square root of 4 or this is the square root of 2 times 2 is just 2 .
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does this simplification leave the square root of 39 alone ?
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in this video , i 'm going to expose you to what is maybe one of at least the top five most useful formulas in mathematics . and if you 've seen many of my videos , you know that i 'm not a big fan of memorizing things . but i will recommend you memorize it with the caveat that you also remember how to prove it , because i do n't want you to just remember things and not know where they came from . but with that said , let me show you what i 'm talking about : it 's the quadratic formula . and as you might guess , it is to solve for the roots , or the zeroes of quadratic equations . so let 's speak in very general terms and i 'll show you some examples . so let 's say i have an equation of the form ax squared plus bx plus c is equal to 0 . you should recognize this . this is a quadratic equation where a , b and c are -- well , a is the coefficient on the x squared term or the second degree term , b is the coefficient on the x term and then c , is , you could imagine , the coefficient on the x to the zero term , or it 's the constant term . now , given that you have a general quadratic equation like this , the quadratic formula tells us that the solutions to this equation are x is equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . and i know it seems crazy and convoluted and hard for you to memorize right now , but as you get a lot more practice you 'll see that it actually is a pretty reasonable formula to stick in your brain someplace . and you might say , gee , this is a wacky formula , where did it come from ? and in the next video i 'm going to show you where it came from . but i want you to get used to using it first . but it really just came from completing the square on this equation right there . if you complete the square here , you 're actually going to get this solution and that is the quadratic formula , right there . so let 's apply it to some problems . let 's start off with something that we could have factored just to verify that it 's giving us the same answer . so let 's say we have x squared plus 4x minus 21 is equal to 0 . so in this situation -- let me do that in a different color -- a is equal to 1 , right ? the coefficient on the x squared term is 1. b is equal to 4 , the coefficient on the x-term . and then c is equal to negative 21 , the constant term . and let 's just plug it in the formula , so what do we get ? we get x , this tells us that x is going to be equal to negative b . negative b is negative 4 -- i put the negative sign in front of that -- negative b plus or minus the square root of b squared . b squared is 16 , right ? 4 squared is 16 , minus 4 times a , which is 1 , times c , which is negative 21 . so we can put a 21 out there and that negative sign will cancel out just like that with that -- since this is the first time we 're doing it , let me not skip too many steps . so negative 21 , just so you can see how it fit in , and then all of that over 2a . a is 1 , so all of that over 2 . so what does this simplify , or hopefully it simplifies ? so we get x is equal to negative 4 plus or minus the square root of -- let 's see we have a negative times a negative , that 's going to give us a positive . and we had 16 plus , let 's see this is 6 , 4 times 1 is 4 times 21 is 84 . 16 plus 84 is 100 . that 's nice . that 's a nice perfect square . all of that over 2 , and so this is going to be equal to negative 4 plus or minus 10 over 2 . we could just divide both of these terms by 2 right now . so this is equal to negative 4 divided by 2 is negative 2 plus or minus 10 divided by 2 is 5 . so that tells us that x could be equal to negative 2 plus 5 , which is 3 , or x could be equal to negative 2 minus 5 , which is negative 7 . so the quadratic formula seems to have given us an answer for this . you can verify just by substituting back in that these do work , or you could even just try to factor this right here . you say what two numbers when you take their product , you get negative 21 and when you take their sum you get positive 4 ? so you 'd get x plus 7 times x minus 3 is equal to negative 21 . notice 7 times negative 3 is negative 21 , 7 minus 3 is positive 4 . you would get x plus -- sorry it 's not negative -- 21 is equal to 0 . there should be a 0 there . so you get x plus 7 is equal to 0 , or x minus 3 is equal to 0 . x could be equal to negative 7 or x could be equal to 3 . so it definitely gives us the same answer as factoring , so you might say , hey why bother with this crazy mess ? and the reason we want to bother with this crazy mess is it 'll also work for problems that are hard to factor . and let 's do a couple of those , let 's do some hard-to-factor problems right now . so let 's scroll down to get some fresh real estate . let 's rewrite the formula again , just in case we have n't had it memorized yet . x is going to be equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . i 'll supply this to another problem . let 's say we have the equation 3x squared plus 6x is equal to negative 10 . well , the first thing we want to do is get it in the form where all of our terms or on the left-hand side , so let 's add 10 to both sides of this equation . we get 3x squared plus the 6x plus 10 is equal to 0 . and now we can use a quadratic formula . so let 's apply it here . so a is equal to 3 . that is a , this is b and this right here is c. so the quadratic formula tells us the solutions to this equation . the roots of this quadratic function , i guess we could call it . x is going to be equal to negative b. b is 6 , so negative 6 plus or minus the square root of b squared . b is 6 , so we get 6 squared minus 4 times a , which is 3 times c , which is 10 . let 's stretch out the radical little bit , all of that over 2 times a , 2 times 3 . so we get x is equal to negative 6 plus or minus the square root of 36 minus -- this is interesting -- minus 4 times 3 times 10 . so this is minus -- 4 times 3 times 10 . so this is minus 120 . all of that over 6 . so this is interesting , you might already realize why it 's interesting . what is this going to simplify to ? 36 minus 120 is what ? that 's 84 . we make this into a 10 , this will become an 11 , this is a 4 . it is 84 , so this is going to be equal to negative 6 plus or minus the square root of -- but not positive 84 , that 's if it 's 120 minus 36 . we have 36 minus 120 . it 's going to be negative 84 all of that 6 . so you might say , gee , this is crazy . what a this silly quadratic formula you 're introducing me to , sal ? it 's worthless . it just gives me a square root of a negative number . it 's not giving me an answer . and the reason why it 's not giving you an answer , at least an answer that you might want , is because this will have no real solutions . in the future , we 're going to introduce something called an imaginary number , which is a square root of a negative number , and then we can actually express this in terms of those numbers . so this actually does have solutions , but they involve imaginary numbers . so this actually has no real solutions , we 're taking the square root of a negative number . so the b squared with the b squared minus 4ac , if this term right here is negative , then you 're not going to have any real solutions . and let 's verify that for ourselves . let 's get our graphic calculator out and let 's graph this equation right here . so , let 's get the graphs that y is equal to -- that 's what i had there before -- 3x squared plus 6x plus 10 . so that 's the equation and we 're going to see where it intersects the x-axis . where does it equal 0 ? so let me graph it . notice , this thing just comes down and then goes back up . its vertex is sitting here above the x-axis and it 's upward-opening . it never intersects the x-axis . so at no point will this expression , will this function , equal 0 . at no point will y equal 0 on this graph . so once again , the quadratic formula seems to be working . let 's do one more example , you can never see enough examples here . and i want to do ones that are , you know , maybe not so obvious to factor . so let 's say we get negative 3x squared plus 12x plus 1 is equal to 0 . now let 's try to do it just having the quadratic formula in our brain . so the x 's that satisfy this equation are going to be negative b . this is b so negative b is negative 12 plus or minus the square root of b squared , of 144 , that 's b squared minus 4 times a , which is negative 3 times c , which is 1 , all of that over 2 times a , over 2 times negative 3 . so all of that over negative 6 , this is going to be equal to negative 12 plus or minus the square root of -- what is this ? it 's a negative times a negative so they cancel out . so i have 144 plus 12 , so that is 156 , right ? 144 plus 12 , all of that over negative 6 . now , i suspect we can simplify this 156 . we could maybe bring some things out of the radical sign . so let 's attempt to do that . so let 's do a prime factorization of 156 . sometimes , this is the hardest part , simplifying the radical . so 156 is the same thing as 2 times 78 . 78 is the same thing as 2 times what ? that 's 2 times 39 . so the square root of 156 is equal to the square root of 2 times 2 times 39 or we could say that 's the square root of 2 times 2 times the square root of 39 . and this , obviously , is just going to be the square root of 4 or this is the square root of 2 times 2 is just 2 . 2 square roots of 39 , if i did that properly , let 's see , 4 times 39 . yeah , it looks like it 's right . so this up here will simplify to negative 12 plus or minus 2 times the square root of 39 , all of that over negative 6 . now we can divide the numerator and the denominator maybe by 2 . so this will be equal to negative 6 plus or minus the square root of 39 over negative 3 . or we could separate these two terms out . we could say this is equal to negative 6 over negative 3 plus or minus the square root of 39 over negative 3 . now , this is just a 2 right here , right ? these cancel out , 6 divided by 3 is 2 , so we get 2 . and now notice , if this is plus and we use this minus sign , the plus will become negative and the negative will become positive . but it still does n't matter , right ? we could say minus or plus , that 's the same thing as plus or minus the square root of 39 nine over 3 . i think that 's about as simple as we can get this answered . i want to make a very clear point of what i did that last step . i did not forget about this negative sign . i just said it does n't matter . it 's going to turn the positive into the negative ; it 's going to turn the negative into the positive . let me rewrite this . so this right here can be rewritten as 2 plus the square root of 39 over negative 3 or 2 minus the square root of 39 over negative 3 , right ? that 's what the plus or minus means , it could be this or that or both of them , really . now in this situation , this negative 3 will turn into 2 minus the square root of 39 over 3 , right ? i 'm just taking this negative out . here the negative and the negative will become a positive , and you get 2 plus the square root of 39 over 3 , right ? a negative times a negative is a positive . so once again , you have 2 plus or minus the square of 39 over 3 . 2 plus or minus the square root of 39 over 3 are solutions to this equation right there . let verify . i 'm just curious what the graph looks like . so let 's just look at it . let me clear this . where is the clear button ? so we have negative 3 three squared plus 12x plus 1 and let 's graph it . let 's see where it intersects the x-axis . it goes up there and then back down again . so 2 plus or minus the square , you see -- the square root of 39 is going to be a little bit more than 6 , right ? because 36 is 6 squared . so it 's going be a little bit more than 6 , so this is going to be a little bit more than 2 . a little bit more than 6 divided by 2 is a little bit more than 2 . so you 're going to get one value that 's a little bit more than 4 and then another value that should be a little bit less than 1 . and that looks like the case , you have 1 , 2 , 3 , 4 . you have a value that 's pretty close to 4 , and then you have another value that is a little bit -- it looks close to 0 but maybe a little bit less than that . so anyway , hopefully you found this application of the quadratic formula helpful .
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we get x , this tells us that x is going to be equal to negative b . negative b is negative 4 -- i put the negative sign in front of that -- negative b plus or minus the square root of b squared . b squared is 16 , right ?
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what does one do when b is a negative number ?
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in this video , i 'm going to expose you to what is maybe one of at least the top five most useful formulas in mathematics . and if you 've seen many of my videos , you know that i 'm not a big fan of memorizing things . but i will recommend you memorize it with the caveat that you also remember how to prove it , because i do n't want you to just remember things and not know where they came from . but with that said , let me show you what i 'm talking about : it 's the quadratic formula . and as you might guess , it is to solve for the roots , or the zeroes of quadratic equations . so let 's speak in very general terms and i 'll show you some examples . so let 's say i have an equation of the form ax squared plus bx plus c is equal to 0 . you should recognize this . this is a quadratic equation where a , b and c are -- well , a is the coefficient on the x squared term or the second degree term , b is the coefficient on the x term and then c , is , you could imagine , the coefficient on the x to the zero term , or it 's the constant term . now , given that you have a general quadratic equation like this , the quadratic formula tells us that the solutions to this equation are x is equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . and i know it seems crazy and convoluted and hard for you to memorize right now , but as you get a lot more practice you 'll see that it actually is a pretty reasonable formula to stick in your brain someplace . and you might say , gee , this is a wacky formula , where did it come from ? and in the next video i 'm going to show you where it came from . but i want you to get used to using it first . but it really just came from completing the square on this equation right there . if you complete the square here , you 're actually going to get this solution and that is the quadratic formula , right there . so let 's apply it to some problems . let 's start off with something that we could have factored just to verify that it 's giving us the same answer . so let 's say we have x squared plus 4x minus 21 is equal to 0 . so in this situation -- let me do that in a different color -- a is equal to 1 , right ? the coefficient on the x squared term is 1. b is equal to 4 , the coefficient on the x-term . and then c is equal to negative 21 , the constant term . and let 's just plug it in the formula , so what do we get ? we get x , this tells us that x is going to be equal to negative b . negative b is negative 4 -- i put the negative sign in front of that -- negative b plus or minus the square root of b squared . b squared is 16 , right ? 4 squared is 16 , minus 4 times a , which is 1 , times c , which is negative 21 . so we can put a 21 out there and that negative sign will cancel out just like that with that -- since this is the first time we 're doing it , let me not skip too many steps . so negative 21 , just so you can see how it fit in , and then all of that over 2a . a is 1 , so all of that over 2 . so what does this simplify , or hopefully it simplifies ? so we get x is equal to negative 4 plus or minus the square root of -- let 's see we have a negative times a negative , that 's going to give us a positive . and we had 16 plus , let 's see this is 6 , 4 times 1 is 4 times 21 is 84 . 16 plus 84 is 100 . that 's nice . that 's a nice perfect square . all of that over 2 , and so this is going to be equal to negative 4 plus or minus 10 over 2 . we could just divide both of these terms by 2 right now . so this is equal to negative 4 divided by 2 is negative 2 plus or minus 10 divided by 2 is 5 . so that tells us that x could be equal to negative 2 plus 5 , which is 3 , or x could be equal to negative 2 minus 5 , which is negative 7 . so the quadratic formula seems to have given us an answer for this . you can verify just by substituting back in that these do work , or you could even just try to factor this right here . you say what two numbers when you take their product , you get negative 21 and when you take their sum you get positive 4 ? so you 'd get x plus 7 times x minus 3 is equal to negative 21 . notice 7 times negative 3 is negative 21 , 7 minus 3 is positive 4 . you would get x plus -- sorry it 's not negative -- 21 is equal to 0 . there should be a 0 there . so you get x plus 7 is equal to 0 , or x minus 3 is equal to 0 . x could be equal to negative 7 or x could be equal to 3 . so it definitely gives us the same answer as factoring , so you might say , hey why bother with this crazy mess ? and the reason we want to bother with this crazy mess is it 'll also work for problems that are hard to factor . and let 's do a couple of those , let 's do some hard-to-factor problems right now . so let 's scroll down to get some fresh real estate . let 's rewrite the formula again , just in case we have n't had it memorized yet . x is going to be equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . i 'll supply this to another problem . let 's say we have the equation 3x squared plus 6x is equal to negative 10 . well , the first thing we want to do is get it in the form where all of our terms or on the left-hand side , so let 's add 10 to both sides of this equation . we get 3x squared plus the 6x plus 10 is equal to 0 . and now we can use a quadratic formula . so let 's apply it here . so a is equal to 3 . that is a , this is b and this right here is c. so the quadratic formula tells us the solutions to this equation . the roots of this quadratic function , i guess we could call it . x is going to be equal to negative b. b is 6 , so negative 6 plus or minus the square root of b squared . b is 6 , so we get 6 squared minus 4 times a , which is 3 times c , which is 10 . let 's stretch out the radical little bit , all of that over 2 times a , 2 times 3 . so we get x is equal to negative 6 plus or minus the square root of 36 minus -- this is interesting -- minus 4 times 3 times 10 . so this is minus -- 4 times 3 times 10 . so this is minus 120 . all of that over 6 . so this is interesting , you might already realize why it 's interesting . what is this going to simplify to ? 36 minus 120 is what ? that 's 84 . we make this into a 10 , this will become an 11 , this is a 4 . it is 84 , so this is going to be equal to negative 6 plus or minus the square root of -- but not positive 84 , that 's if it 's 120 minus 36 . we have 36 minus 120 . it 's going to be negative 84 all of that 6 . so you might say , gee , this is crazy . what a this silly quadratic formula you 're introducing me to , sal ? it 's worthless . it just gives me a square root of a negative number . it 's not giving me an answer . and the reason why it 's not giving you an answer , at least an answer that you might want , is because this will have no real solutions . in the future , we 're going to introduce something called an imaginary number , which is a square root of a negative number , and then we can actually express this in terms of those numbers . so this actually does have solutions , but they involve imaginary numbers . so this actually has no real solutions , we 're taking the square root of a negative number . so the b squared with the b squared minus 4ac , if this term right here is negative , then you 're not going to have any real solutions . and let 's verify that for ourselves . let 's get our graphic calculator out and let 's graph this equation right here . so , let 's get the graphs that y is equal to -- that 's what i had there before -- 3x squared plus 6x plus 10 . so that 's the equation and we 're going to see where it intersects the x-axis . where does it equal 0 ? so let me graph it . notice , this thing just comes down and then goes back up . its vertex is sitting here above the x-axis and it 's upward-opening . it never intersects the x-axis . so at no point will this expression , will this function , equal 0 . at no point will y equal 0 on this graph . so once again , the quadratic formula seems to be working . let 's do one more example , you can never see enough examples here . and i want to do ones that are , you know , maybe not so obvious to factor . so let 's say we get negative 3x squared plus 12x plus 1 is equal to 0 . now let 's try to do it just having the quadratic formula in our brain . so the x 's that satisfy this equation are going to be negative b . this is b so negative b is negative 12 plus or minus the square root of b squared , of 144 , that 's b squared minus 4 times a , which is negative 3 times c , which is 1 , all of that over 2 times a , over 2 times negative 3 . so all of that over negative 6 , this is going to be equal to negative 12 plus or minus the square root of -- what is this ? it 's a negative times a negative so they cancel out . so i have 144 plus 12 , so that is 156 , right ? 144 plus 12 , all of that over negative 6 . now , i suspect we can simplify this 156 . we could maybe bring some things out of the radical sign . so let 's attempt to do that . so let 's do a prime factorization of 156 . sometimes , this is the hardest part , simplifying the radical . so 156 is the same thing as 2 times 78 . 78 is the same thing as 2 times what ? that 's 2 times 39 . so the square root of 156 is equal to the square root of 2 times 2 times 39 or we could say that 's the square root of 2 times 2 times the square root of 39 . and this , obviously , is just going to be the square root of 4 or this is the square root of 2 times 2 is just 2 . 2 square roots of 39 , if i did that properly , let 's see , 4 times 39 . yeah , it looks like it 's right . so this up here will simplify to negative 12 plus or minus 2 times the square root of 39 , all of that over negative 6 . now we can divide the numerator and the denominator maybe by 2 . so this will be equal to negative 6 plus or minus the square root of 39 over negative 3 . or we could separate these two terms out . we could say this is equal to negative 6 over negative 3 plus or minus the square root of 39 over negative 3 . now , this is just a 2 right here , right ? these cancel out , 6 divided by 3 is 2 , so we get 2 . and now notice , if this is plus and we use this minus sign , the plus will become negative and the negative will become positive . but it still does n't matter , right ? we could say minus or plus , that 's the same thing as plus or minus the square root of 39 nine over 3 . i think that 's about as simple as we can get this answered . i want to make a very clear point of what i did that last step . i did not forget about this negative sign . i just said it does n't matter . it 's going to turn the positive into the negative ; it 's going to turn the negative into the positive . let me rewrite this . so this right here can be rewritten as 2 plus the square root of 39 over negative 3 or 2 minus the square root of 39 over negative 3 , right ? that 's what the plus or minus means , it could be this or that or both of them , really . now in this situation , this negative 3 will turn into 2 minus the square root of 39 over 3 , right ? i 'm just taking this negative out . here the negative and the negative will become a positive , and you get 2 plus the square root of 39 over 3 , right ? a negative times a negative is a positive . so once again , you have 2 plus or minus the square of 39 over 3 . 2 plus or minus the square root of 39 over 3 are solutions to this equation right there . let verify . i 'm just curious what the graph looks like . so let 's just look at it . let me clear this . where is the clear button ? so we have negative 3 three squared plus 12x plus 1 and let 's graph it . let 's see where it intersects the x-axis . it goes up there and then back down again . so 2 plus or minus the square , you see -- the square root of 39 is going to be a little bit more than 6 , right ? because 36 is 6 squared . so it 's going be a little bit more than 6 , so this is going to be a little bit more than 2 . a little bit more than 6 divided by 2 is a little bit more than 2 . so you 're going to get one value that 's a little bit more than 4 and then another value that should be a little bit less than 1 . and that looks like the case , you have 1 , 2 , 3 , 4 . you have a value that 's pretty close to 4 , and then you have another value that is a little bit -- it looks close to 0 but maybe a little bit less than that . so anyway , hopefully you found this application of the quadratic formula helpful .
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this is a quadratic equation where a , b and c are -- well , a is the coefficient on the x squared term or the second degree term , b is the coefficient on the x term and then c , is , you could imagine , the coefficient on the x to the zero term , or it 's the constant term . now , given that you have a general quadratic equation like this , the quadratic formula tells us that the solutions to this equation are x is equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . and i know it seems crazy and convoluted and hard for you to memorize right now , but as you get a lot more practice you 'll see that it actually is a pretty reasonable formula to stick in your brain someplace .
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how would you put the quadratic equation be put into vertex form ?
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in this video , i 'm going to expose you to what is maybe one of at least the top five most useful formulas in mathematics . and if you 've seen many of my videos , you know that i 'm not a big fan of memorizing things . but i will recommend you memorize it with the caveat that you also remember how to prove it , because i do n't want you to just remember things and not know where they came from . but with that said , let me show you what i 'm talking about : it 's the quadratic formula . and as you might guess , it is to solve for the roots , or the zeroes of quadratic equations . so let 's speak in very general terms and i 'll show you some examples . so let 's say i have an equation of the form ax squared plus bx plus c is equal to 0 . you should recognize this . this is a quadratic equation where a , b and c are -- well , a is the coefficient on the x squared term or the second degree term , b is the coefficient on the x term and then c , is , you could imagine , the coefficient on the x to the zero term , or it 's the constant term . now , given that you have a general quadratic equation like this , the quadratic formula tells us that the solutions to this equation are x is equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . and i know it seems crazy and convoluted and hard for you to memorize right now , but as you get a lot more practice you 'll see that it actually is a pretty reasonable formula to stick in your brain someplace . and you might say , gee , this is a wacky formula , where did it come from ? and in the next video i 'm going to show you where it came from . but i want you to get used to using it first . but it really just came from completing the square on this equation right there . if you complete the square here , you 're actually going to get this solution and that is the quadratic formula , right there . so let 's apply it to some problems . let 's start off with something that we could have factored just to verify that it 's giving us the same answer . so let 's say we have x squared plus 4x minus 21 is equal to 0 . so in this situation -- let me do that in a different color -- a is equal to 1 , right ? the coefficient on the x squared term is 1. b is equal to 4 , the coefficient on the x-term . and then c is equal to negative 21 , the constant term . and let 's just plug it in the formula , so what do we get ? we get x , this tells us that x is going to be equal to negative b . negative b is negative 4 -- i put the negative sign in front of that -- negative b plus or minus the square root of b squared . b squared is 16 , right ? 4 squared is 16 , minus 4 times a , which is 1 , times c , which is negative 21 . so we can put a 21 out there and that negative sign will cancel out just like that with that -- since this is the first time we 're doing it , let me not skip too many steps . so negative 21 , just so you can see how it fit in , and then all of that over 2a . a is 1 , so all of that over 2 . so what does this simplify , or hopefully it simplifies ? so we get x is equal to negative 4 plus or minus the square root of -- let 's see we have a negative times a negative , that 's going to give us a positive . and we had 16 plus , let 's see this is 6 , 4 times 1 is 4 times 21 is 84 . 16 plus 84 is 100 . that 's nice . that 's a nice perfect square . all of that over 2 , and so this is going to be equal to negative 4 plus or minus 10 over 2 . we could just divide both of these terms by 2 right now . so this is equal to negative 4 divided by 2 is negative 2 plus or minus 10 divided by 2 is 5 . so that tells us that x could be equal to negative 2 plus 5 , which is 3 , or x could be equal to negative 2 minus 5 , which is negative 7 . so the quadratic formula seems to have given us an answer for this . you can verify just by substituting back in that these do work , or you could even just try to factor this right here . you say what two numbers when you take their product , you get negative 21 and when you take their sum you get positive 4 ? so you 'd get x plus 7 times x minus 3 is equal to negative 21 . notice 7 times negative 3 is negative 21 , 7 minus 3 is positive 4 . you would get x plus -- sorry it 's not negative -- 21 is equal to 0 . there should be a 0 there . so you get x plus 7 is equal to 0 , or x minus 3 is equal to 0 . x could be equal to negative 7 or x could be equal to 3 . so it definitely gives us the same answer as factoring , so you might say , hey why bother with this crazy mess ? and the reason we want to bother with this crazy mess is it 'll also work for problems that are hard to factor . and let 's do a couple of those , let 's do some hard-to-factor problems right now . so let 's scroll down to get some fresh real estate . let 's rewrite the formula again , just in case we have n't had it memorized yet . x is going to be equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . i 'll supply this to another problem . let 's say we have the equation 3x squared plus 6x is equal to negative 10 . well , the first thing we want to do is get it in the form where all of our terms or on the left-hand side , so let 's add 10 to both sides of this equation . we get 3x squared plus the 6x plus 10 is equal to 0 . and now we can use a quadratic formula . so let 's apply it here . so a is equal to 3 . that is a , this is b and this right here is c. so the quadratic formula tells us the solutions to this equation . the roots of this quadratic function , i guess we could call it . x is going to be equal to negative b. b is 6 , so negative 6 plus or minus the square root of b squared . b is 6 , so we get 6 squared minus 4 times a , which is 3 times c , which is 10 . let 's stretch out the radical little bit , all of that over 2 times a , 2 times 3 . so we get x is equal to negative 6 plus or minus the square root of 36 minus -- this is interesting -- minus 4 times 3 times 10 . so this is minus -- 4 times 3 times 10 . so this is minus 120 . all of that over 6 . so this is interesting , you might already realize why it 's interesting . what is this going to simplify to ? 36 minus 120 is what ? that 's 84 . we make this into a 10 , this will become an 11 , this is a 4 . it is 84 , so this is going to be equal to negative 6 plus or minus the square root of -- but not positive 84 , that 's if it 's 120 minus 36 . we have 36 minus 120 . it 's going to be negative 84 all of that 6 . so you might say , gee , this is crazy . what a this silly quadratic formula you 're introducing me to , sal ? it 's worthless . it just gives me a square root of a negative number . it 's not giving me an answer . and the reason why it 's not giving you an answer , at least an answer that you might want , is because this will have no real solutions . in the future , we 're going to introduce something called an imaginary number , which is a square root of a negative number , and then we can actually express this in terms of those numbers . so this actually does have solutions , but they involve imaginary numbers . so this actually has no real solutions , we 're taking the square root of a negative number . so the b squared with the b squared minus 4ac , if this term right here is negative , then you 're not going to have any real solutions . and let 's verify that for ourselves . let 's get our graphic calculator out and let 's graph this equation right here . so , let 's get the graphs that y is equal to -- that 's what i had there before -- 3x squared plus 6x plus 10 . so that 's the equation and we 're going to see where it intersects the x-axis . where does it equal 0 ? so let me graph it . notice , this thing just comes down and then goes back up . its vertex is sitting here above the x-axis and it 's upward-opening . it never intersects the x-axis . so at no point will this expression , will this function , equal 0 . at no point will y equal 0 on this graph . so once again , the quadratic formula seems to be working . let 's do one more example , you can never see enough examples here . and i want to do ones that are , you know , maybe not so obvious to factor . so let 's say we get negative 3x squared plus 12x plus 1 is equal to 0 . now let 's try to do it just having the quadratic formula in our brain . so the x 's that satisfy this equation are going to be negative b . this is b so negative b is negative 12 plus or minus the square root of b squared , of 144 , that 's b squared minus 4 times a , which is negative 3 times c , which is 1 , all of that over 2 times a , over 2 times negative 3 . so all of that over negative 6 , this is going to be equal to negative 12 plus or minus the square root of -- what is this ? it 's a negative times a negative so they cancel out . so i have 144 plus 12 , so that is 156 , right ? 144 plus 12 , all of that over negative 6 . now , i suspect we can simplify this 156 . we could maybe bring some things out of the radical sign . so let 's attempt to do that . so let 's do a prime factorization of 156 . sometimes , this is the hardest part , simplifying the radical . so 156 is the same thing as 2 times 78 . 78 is the same thing as 2 times what ? that 's 2 times 39 . so the square root of 156 is equal to the square root of 2 times 2 times 39 or we could say that 's the square root of 2 times 2 times the square root of 39 . and this , obviously , is just going to be the square root of 4 or this is the square root of 2 times 2 is just 2 . 2 square roots of 39 , if i did that properly , let 's see , 4 times 39 . yeah , it looks like it 's right . so this up here will simplify to negative 12 plus or minus 2 times the square root of 39 , all of that over negative 6 . now we can divide the numerator and the denominator maybe by 2 . so this will be equal to negative 6 plus or minus the square root of 39 over negative 3 . or we could separate these two terms out . we could say this is equal to negative 6 over negative 3 plus or minus the square root of 39 over negative 3 . now , this is just a 2 right here , right ? these cancel out , 6 divided by 3 is 2 , so we get 2 . and now notice , if this is plus and we use this minus sign , the plus will become negative and the negative will become positive . but it still does n't matter , right ? we could say minus or plus , that 's the same thing as plus or minus the square root of 39 nine over 3 . i think that 's about as simple as we can get this answered . i want to make a very clear point of what i did that last step . i did not forget about this negative sign . i just said it does n't matter . it 's going to turn the positive into the negative ; it 's going to turn the negative into the positive . let me rewrite this . so this right here can be rewritten as 2 plus the square root of 39 over negative 3 or 2 minus the square root of 39 over negative 3 , right ? that 's what the plus or minus means , it could be this or that or both of them , really . now in this situation , this negative 3 will turn into 2 minus the square root of 39 over 3 , right ? i 'm just taking this negative out . here the negative and the negative will become a positive , and you get 2 plus the square root of 39 over 3 , right ? a negative times a negative is a positive . so once again , you have 2 plus or minus the square of 39 over 3 . 2 plus or minus the square root of 39 over 3 are solutions to this equation right there . let verify . i 'm just curious what the graph looks like . so let 's just look at it . let me clear this . where is the clear button ? so we have negative 3 three squared plus 12x plus 1 and let 's graph it . let 's see where it intersects the x-axis . it goes up there and then back down again . so 2 plus or minus the square , you see -- the square root of 39 is going to be a little bit more than 6 , right ? because 36 is 6 squared . so it 's going be a little bit more than 6 , so this is going to be a little bit more than 2 . a little bit more than 6 divided by 2 is a little bit more than 2 . so you 're going to get one value that 's a little bit more than 4 and then another value that should be a little bit less than 1 . and that looks like the case , you have 1 , 2 , 3 , 4 . you have a value that 's pretty close to 4 , and then you have another value that is a little bit -- it looks close to 0 but maybe a little bit less than that . so anyway , hopefully you found this application of the quadratic formula helpful .
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this is a quadratic equation where a , b and c are -- well , a is the coefficient on the x squared term or the second degree term , b is the coefficient on the x term and then c , is , you could imagine , the coefficient on the x to the zero term , or it 's the constant term . now , given that you have a general quadratic equation like this , the quadratic formula tells us that the solutions to this equation are x is equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . and i know it seems crazy and convoluted and hard for you to memorize right now , but as you get a lot more practice you 'll see that it actually is a pretty reasonable formula to stick in your brain someplace .
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also how could an equation in vertex form help you graph a quadratic equation ?
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in this video , i 'm going to expose you to what is maybe one of at least the top five most useful formulas in mathematics . and if you 've seen many of my videos , you know that i 'm not a big fan of memorizing things . but i will recommend you memorize it with the caveat that you also remember how to prove it , because i do n't want you to just remember things and not know where they came from . but with that said , let me show you what i 'm talking about : it 's the quadratic formula . and as you might guess , it is to solve for the roots , or the zeroes of quadratic equations . so let 's speak in very general terms and i 'll show you some examples . so let 's say i have an equation of the form ax squared plus bx plus c is equal to 0 . you should recognize this . this is a quadratic equation where a , b and c are -- well , a is the coefficient on the x squared term or the second degree term , b is the coefficient on the x term and then c , is , you could imagine , the coefficient on the x to the zero term , or it 's the constant term . now , given that you have a general quadratic equation like this , the quadratic formula tells us that the solutions to this equation are x is equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . and i know it seems crazy and convoluted and hard for you to memorize right now , but as you get a lot more practice you 'll see that it actually is a pretty reasonable formula to stick in your brain someplace . and you might say , gee , this is a wacky formula , where did it come from ? and in the next video i 'm going to show you where it came from . but i want you to get used to using it first . but it really just came from completing the square on this equation right there . if you complete the square here , you 're actually going to get this solution and that is the quadratic formula , right there . so let 's apply it to some problems . let 's start off with something that we could have factored just to verify that it 's giving us the same answer . so let 's say we have x squared plus 4x minus 21 is equal to 0 . so in this situation -- let me do that in a different color -- a is equal to 1 , right ? the coefficient on the x squared term is 1. b is equal to 4 , the coefficient on the x-term . and then c is equal to negative 21 , the constant term . and let 's just plug it in the formula , so what do we get ? we get x , this tells us that x is going to be equal to negative b . negative b is negative 4 -- i put the negative sign in front of that -- negative b plus or minus the square root of b squared . b squared is 16 , right ? 4 squared is 16 , minus 4 times a , which is 1 , times c , which is negative 21 . so we can put a 21 out there and that negative sign will cancel out just like that with that -- since this is the first time we 're doing it , let me not skip too many steps . so negative 21 , just so you can see how it fit in , and then all of that over 2a . a is 1 , so all of that over 2 . so what does this simplify , or hopefully it simplifies ? so we get x is equal to negative 4 plus or minus the square root of -- let 's see we have a negative times a negative , that 's going to give us a positive . and we had 16 plus , let 's see this is 6 , 4 times 1 is 4 times 21 is 84 . 16 plus 84 is 100 . that 's nice . that 's a nice perfect square . all of that over 2 , and so this is going to be equal to negative 4 plus or minus 10 over 2 . we could just divide both of these terms by 2 right now . so this is equal to negative 4 divided by 2 is negative 2 plus or minus 10 divided by 2 is 5 . so that tells us that x could be equal to negative 2 plus 5 , which is 3 , or x could be equal to negative 2 minus 5 , which is negative 7 . so the quadratic formula seems to have given us an answer for this . you can verify just by substituting back in that these do work , or you could even just try to factor this right here . you say what two numbers when you take their product , you get negative 21 and when you take their sum you get positive 4 ? so you 'd get x plus 7 times x minus 3 is equal to negative 21 . notice 7 times negative 3 is negative 21 , 7 minus 3 is positive 4 . you would get x plus -- sorry it 's not negative -- 21 is equal to 0 . there should be a 0 there . so you get x plus 7 is equal to 0 , or x minus 3 is equal to 0 . x could be equal to negative 7 or x could be equal to 3 . so it definitely gives us the same answer as factoring , so you might say , hey why bother with this crazy mess ? and the reason we want to bother with this crazy mess is it 'll also work for problems that are hard to factor . and let 's do a couple of those , let 's do some hard-to-factor problems right now . so let 's scroll down to get some fresh real estate . let 's rewrite the formula again , just in case we have n't had it memorized yet . x is going to be equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . i 'll supply this to another problem . let 's say we have the equation 3x squared plus 6x is equal to negative 10 . well , the first thing we want to do is get it in the form where all of our terms or on the left-hand side , so let 's add 10 to both sides of this equation . we get 3x squared plus the 6x plus 10 is equal to 0 . and now we can use a quadratic formula . so let 's apply it here . so a is equal to 3 . that is a , this is b and this right here is c. so the quadratic formula tells us the solutions to this equation . the roots of this quadratic function , i guess we could call it . x is going to be equal to negative b. b is 6 , so negative 6 plus or minus the square root of b squared . b is 6 , so we get 6 squared minus 4 times a , which is 3 times c , which is 10 . let 's stretch out the radical little bit , all of that over 2 times a , 2 times 3 . so we get x is equal to negative 6 plus or minus the square root of 36 minus -- this is interesting -- minus 4 times 3 times 10 . so this is minus -- 4 times 3 times 10 . so this is minus 120 . all of that over 6 . so this is interesting , you might already realize why it 's interesting . what is this going to simplify to ? 36 minus 120 is what ? that 's 84 . we make this into a 10 , this will become an 11 , this is a 4 . it is 84 , so this is going to be equal to negative 6 plus or minus the square root of -- but not positive 84 , that 's if it 's 120 minus 36 . we have 36 minus 120 . it 's going to be negative 84 all of that 6 . so you might say , gee , this is crazy . what a this silly quadratic formula you 're introducing me to , sal ? it 's worthless . it just gives me a square root of a negative number . it 's not giving me an answer . and the reason why it 's not giving you an answer , at least an answer that you might want , is because this will have no real solutions . in the future , we 're going to introduce something called an imaginary number , which is a square root of a negative number , and then we can actually express this in terms of those numbers . so this actually does have solutions , but they involve imaginary numbers . so this actually has no real solutions , we 're taking the square root of a negative number . so the b squared with the b squared minus 4ac , if this term right here is negative , then you 're not going to have any real solutions . and let 's verify that for ourselves . let 's get our graphic calculator out and let 's graph this equation right here . so , let 's get the graphs that y is equal to -- that 's what i had there before -- 3x squared plus 6x plus 10 . so that 's the equation and we 're going to see where it intersects the x-axis . where does it equal 0 ? so let me graph it . notice , this thing just comes down and then goes back up . its vertex is sitting here above the x-axis and it 's upward-opening . it never intersects the x-axis . so at no point will this expression , will this function , equal 0 . at no point will y equal 0 on this graph . so once again , the quadratic formula seems to be working . let 's do one more example , you can never see enough examples here . and i want to do ones that are , you know , maybe not so obvious to factor . so let 's say we get negative 3x squared plus 12x plus 1 is equal to 0 . now let 's try to do it just having the quadratic formula in our brain . so the x 's that satisfy this equation are going to be negative b . this is b so negative b is negative 12 plus or minus the square root of b squared , of 144 , that 's b squared minus 4 times a , which is negative 3 times c , which is 1 , all of that over 2 times a , over 2 times negative 3 . so all of that over negative 6 , this is going to be equal to negative 12 plus or minus the square root of -- what is this ? it 's a negative times a negative so they cancel out . so i have 144 plus 12 , so that is 156 , right ? 144 plus 12 , all of that over negative 6 . now , i suspect we can simplify this 156 . we could maybe bring some things out of the radical sign . so let 's attempt to do that . so let 's do a prime factorization of 156 . sometimes , this is the hardest part , simplifying the radical . so 156 is the same thing as 2 times 78 . 78 is the same thing as 2 times what ? that 's 2 times 39 . so the square root of 156 is equal to the square root of 2 times 2 times 39 or we could say that 's the square root of 2 times 2 times the square root of 39 . and this , obviously , is just going to be the square root of 4 or this is the square root of 2 times 2 is just 2 . 2 square roots of 39 , if i did that properly , let 's see , 4 times 39 . yeah , it looks like it 's right . so this up here will simplify to negative 12 plus or minus 2 times the square root of 39 , all of that over negative 6 . now we can divide the numerator and the denominator maybe by 2 . so this will be equal to negative 6 plus or minus the square root of 39 over negative 3 . or we could separate these two terms out . we could say this is equal to negative 6 over negative 3 plus or minus the square root of 39 over negative 3 . now , this is just a 2 right here , right ? these cancel out , 6 divided by 3 is 2 , so we get 2 . and now notice , if this is plus and we use this minus sign , the plus will become negative and the negative will become positive . but it still does n't matter , right ? we could say minus or plus , that 's the same thing as plus or minus the square root of 39 nine over 3 . i think that 's about as simple as we can get this answered . i want to make a very clear point of what i did that last step . i did not forget about this negative sign . i just said it does n't matter . it 's going to turn the positive into the negative ; it 's going to turn the negative into the positive . let me rewrite this . so this right here can be rewritten as 2 plus the square root of 39 over negative 3 or 2 minus the square root of 39 over negative 3 , right ? that 's what the plus or minus means , it could be this or that or both of them , really . now in this situation , this negative 3 will turn into 2 minus the square root of 39 over 3 , right ? i 'm just taking this negative out . here the negative and the negative will become a positive , and you get 2 plus the square root of 39 over 3 , right ? a negative times a negative is a positive . so once again , you have 2 plus or minus the square of 39 over 3 . 2 plus or minus the square root of 39 over 3 are solutions to this equation right there . let verify . i 'm just curious what the graph looks like . so let 's just look at it . let me clear this . where is the clear button ? so we have negative 3 three squared plus 12x plus 1 and let 's graph it . let 's see where it intersects the x-axis . it goes up there and then back down again . so 2 plus or minus the square , you see -- the square root of 39 is going to be a little bit more than 6 , right ? because 36 is 6 squared . so it 's going be a little bit more than 6 , so this is going to be a little bit more than 2 . a little bit more than 6 divided by 2 is a little bit more than 2 . so you 're going to get one value that 's a little bit more than 4 and then another value that should be a little bit less than 1 . and that looks like the case , you have 1 , 2 , 3 , 4 . you have a value that 's pretty close to 4 , and then you have another value that is a little bit -- it looks close to 0 but maybe a little bit less than that . so anyway , hopefully you found this application of the quadratic formula helpful .
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we make this into a 10 , this will become an 11 , this is a 4 . it is 84 , so this is going to be equal to negative 6 plus or minus the square root of -- but not positive 84 , that 's if it 's 120 minus 36 . we have 36 minus 120 .
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why can we not square the negative root thereby getting ( 36 +- 84 ) /36 ?
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in this video , i 'm going to expose you to what is maybe one of at least the top five most useful formulas in mathematics . and if you 've seen many of my videos , you know that i 'm not a big fan of memorizing things . but i will recommend you memorize it with the caveat that you also remember how to prove it , because i do n't want you to just remember things and not know where they came from . but with that said , let me show you what i 'm talking about : it 's the quadratic formula . and as you might guess , it is to solve for the roots , or the zeroes of quadratic equations . so let 's speak in very general terms and i 'll show you some examples . so let 's say i have an equation of the form ax squared plus bx plus c is equal to 0 . you should recognize this . this is a quadratic equation where a , b and c are -- well , a is the coefficient on the x squared term or the second degree term , b is the coefficient on the x term and then c , is , you could imagine , the coefficient on the x to the zero term , or it 's the constant term . now , given that you have a general quadratic equation like this , the quadratic formula tells us that the solutions to this equation are x is equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . and i know it seems crazy and convoluted and hard for you to memorize right now , but as you get a lot more practice you 'll see that it actually is a pretty reasonable formula to stick in your brain someplace . and you might say , gee , this is a wacky formula , where did it come from ? and in the next video i 'm going to show you where it came from . but i want you to get used to using it first . but it really just came from completing the square on this equation right there . if you complete the square here , you 're actually going to get this solution and that is the quadratic formula , right there . so let 's apply it to some problems . let 's start off with something that we could have factored just to verify that it 's giving us the same answer . so let 's say we have x squared plus 4x minus 21 is equal to 0 . so in this situation -- let me do that in a different color -- a is equal to 1 , right ? the coefficient on the x squared term is 1. b is equal to 4 , the coefficient on the x-term . and then c is equal to negative 21 , the constant term . and let 's just plug it in the formula , so what do we get ? we get x , this tells us that x is going to be equal to negative b . negative b is negative 4 -- i put the negative sign in front of that -- negative b plus or minus the square root of b squared . b squared is 16 , right ? 4 squared is 16 , minus 4 times a , which is 1 , times c , which is negative 21 . so we can put a 21 out there and that negative sign will cancel out just like that with that -- since this is the first time we 're doing it , let me not skip too many steps . so negative 21 , just so you can see how it fit in , and then all of that over 2a . a is 1 , so all of that over 2 . so what does this simplify , or hopefully it simplifies ? so we get x is equal to negative 4 plus or minus the square root of -- let 's see we have a negative times a negative , that 's going to give us a positive . and we had 16 plus , let 's see this is 6 , 4 times 1 is 4 times 21 is 84 . 16 plus 84 is 100 . that 's nice . that 's a nice perfect square . all of that over 2 , and so this is going to be equal to negative 4 plus or minus 10 over 2 . we could just divide both of these terms by 2 right now . so this is equal to negative 4 divided by 2 is negative 2 plus or minus 10 divided by 2 is 5 . so that tells us that x could be equal to negative 2 plus 5 , which is 3 , or x could be equal to negative 2 minus 5 , which is negative 7 . so the quadratic formula seems to have given us an answer for this . you can verify just by substituting back in that these do work , or you could even just try to factor this right here . you say what two numbers when you take their product , you get negative 21 and when you take their sum you get positive 4 ? so you 'd get x plus 7 times x minus 3 is equal to negative 21 . notice 7 times negative 3 is negative 21 , 7 minus 3 is positive 4 . you would get x plus -- sorry it 's not negative -- 21 is equal to 0 . there should be a 0 there . so you get x plus 7 is equal to 0 , or x minus 3 is equal to 0 . x could be equal to negative 7 or x could be equal to 3 . so it definitely gives us the same answer as factoring , so you might say , hey why bother with this crazy mess ? and the reason we want to bother with this crazy mess is it 'll also work for problems that are hard to factor . and let 's do a couple of those , let 's do some hard-to-factor problems right now . so let 's scroll down to get some fresh real estate . let 's rewrite the formula again , just in case we have n't had it memorized yet . x is going to be equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . i 'll supply this to another problem . let 's say we have the equation 3x squared plus 6x is equal to negative 10 . well , the first thing we want to do is get it in the form where all of our terms or on the left-hand side , so let 's add 10 to both sides of this equation . we get 3x squared plus the 6x plus 10 is equal to 0 . and now we can use a quadratic formula . so let 's apply it here . so a is equal to 3 . that is a , this is b and this right here is c. so the quadratic formula tells us the solutions to this equation . the roots of this quadratic function , i guess we could call it . x is going to be equal to negative b. b is 6 , so negative 6 plus or minus the square root of b squared . b is 6 , so we get 6 squared minus 4 times a , which is 3 times c , which is 10 . let 's stretch out the radical little bit , all of that over 2 times a , 2 times 3 . so we get x is equal to negative 6 plus or minus the square root of 36 minus -- this is interesting -- minus 4 times 3 times 10 . so this is minus -- 4 times 3 times 10 . so this is minus 120 . all of that over 6 . so this is interesting , you might already realize why it 's interesting . what is this going to simplify to ? 36 minus 120 is what ? that 's 84 . we make this into a 10 , this will become an 11 , this is a 4 . it is 84 , so this is going to be equal to negative 6 plus or minus the square root of -- but not positive 84 , that 's if it 's 120 minus 36 . we have 36 minus 120 . it 's going to be negative 84 all of that 6 . so you might say , gee , this is crazy . what a this silly quadratic formula you 're introducing me to , sal ? it 's worthless . it just gives me a square root of a negative number . it 's not giving me an answer . and the reason why it 's not giving you an answer , at least an answer that you might want , is because this will have no real solutions . in the future , we 're going to introduce something called an imaginary number , which is a square root of a negative number , and then we can actually express this in terms of those numbers . so this actually does have solutions , but they involve imaginary numbers . so this actually has no real solutions , we 're taking the square root of a negative number . so the b squared with the b squared minus 4ac , if this term right here is negative , then you 're not going to have any real solutions . and let 's verify that for ourselves . let 's get our graphic calculator out and let 's graph this equation right here . so , let 's get the graphs that y is equal to -- that 's what i had there before -- 3x squared plus 6x plus 10 . so that 's the equation and we 're going to see where it intersects the x-axis . where does it equal 0 ? so let me graph it . notice , this thing just comes down and then goes back up . its vertex is sitting here above the x-axis and it 's upward-opening . it never intersects the x-axis . so at no point will this expression , will this function , equal 0 . at no point will y equal 0 on this graph . so once again , the quadratic formula seems to be working . let 's do one more example , you can never see enough examples here . and i want to do ones that are , you know , maybe not so obvious to factor . so let 's say we get negative 3x squared plus 12x plus 1 is equal to 0 . now let 's try to do it just having the quadratic formula in our brain . so the x 's that satisfy this equation are going to be negative b . this is b so negative b is negative 12 plus or minus the square root of b squared , of 144 , that 's b squared minus 4 times a , which is negative 3 times c , which is 1 , all of that over 2 times a , over 2 times negative 3 . so all of that over negative 6 , this is going to be equal to negative 12 plus or minus the square root of -- what is this ? it 's a negative times a negative so they cancel out . so i have 144 plus 12 , so that is 156 , right ? 144 plus 12 , all of that over negative 6 . now , i suspect we can simplify this 156 . we could maybe bring some things out of the radical sign . so let 's attempt to do that . so let 's do a prime factorization of 156 . sometimes , this is the hardest part , simplifying the radical . so 156 is the same thing as 2 times 78 . 78 is the same thing as 2 times what ? that 's 2 times 39 . so the square root of 156 is equal to the square root of 2 times 2 times 39 or we could say that 's the square root of 2 times 2 times the square root of 39 . and this , obviously , is just going to be the square root of 4 or this is the square root of 2 times 2 is just 2 . 2 square roots of 39 , if i did that properly , let 's see , 4 times 39 . yeah , it looks like it 's right . so this up here will simplify to negative 12 plus or minus 2 times the square root of 39 , all of that over negative 6 . now we can divide the numerator and the denominator maybe by 2 . so this will be equal to negative 6 plus or minus the square root of 39 over negative 3 . or we could separate these two terms out . we could say this is equal to negative 6 over negative 3 plus or minus the square root of 39 over negative 3 . now , this is just a 2 right here , right ? these cancel out , 6 divided by 3 is 2 , so we get 2 . and now notice , if this is plus and we use this minus sign , the plus will become negative and the negative will become positive . but it still does n't matter , right ? we could say minus or plus , that 's the same thing as plus or minus the square root of 39 nine over 3 . i think that 's about as simple as we can get this answered . i want to make a very clear point of what i did that last step . i did not forget about this negative sign . i just said it does n't matter . it 's going to turn the positive into the negative ; it 's going to turn the negative into the positive . let me rewrite this . so this right here can be rewritten as 2 plus the square root of 39 over negative 3 or 2 minus the square root of 39 over negative 3 , right ? that 's what the plus or minus means , it could be this or that or both of them , really . now in this situation , this negative 3 will turn into 2 minus the square root of 39 over 3 , right ? i 'm just taking this negative out . here the negative and the negative will become a positive , and you get 2 plus the square root of 39 over 3 , right ? a negative times a negative is a positive . so once again , you have 2 plus or minus the square of 39 over 3 . 2 plus or minus the square root of 39 over 3 are solutions to this equation right there . let verify . i 'm just curious what the graph looks like . so let 's just look at it . let me clear this . where is the clear button ? so we have negative 3 three squared plus 12x plus 1 and let 's graph it . let 's see where it intersects the x-axis . it goes up there and then back down again . so 2 plus or minus the square , you see -- the square root of 39 is going to be a little bit more than 6 , right ? because 36 is 6 squared . so it 's going be a little bit more than 6 , so this is going to be a little bit more than 2 . a little bit more than 6 divided by 2 is a little bit more than 2 . so you 're going to get one value that 's a little bit more than 4 and then another value that should be a little bit less than 1 . and that looks like the case , you have 1 , 2 , 3 , 4 . you have a value that 's pretty close to 4 , and then you have another value that is a little bit -- it looks close to 0 but maybe a little bit less than that . so anyway , hopefully you found this application of the quadratic formula helpful .
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that 's 2 times 39 . so the square root of 156 is equal to the square root of 2 times 2 times 39 or we could say that 's the square root of 2 times 2 times the square root of 39 . and this , obviously , is just going to be the square root of 4 or this is the square root of 2 times 2 is just 2 .
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what is the point in squaring the terms before find the square root ?
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in this video , i 'm going to expose you to what is maybe one of at least the top five most useful formulas in mathematics . and if you 've seen many of my videos , you know that i 'm not a big fan of memorizing things . but i will recommend you memorize it with the caveat that you also remember how to prove it , because i do n't want you to just remember things and not know where they came from . but with that said , let me show you what i 'm talking about : it 's the quadratic formula . and as you might guess , it is to solve for the roots , or the zeroes of quadratic equations . so let 's speak in very general terms and i 'll show you some examples . so let 's say i have an equation of the form ax squared plus bx plus c is equal to 0 . you should recognize this . this is a quadratic equation where a , b and c are -- well , a is the coefficient on the x squared term or the second degree term , b is the coefficient on the x term and then c , is , you could imagine , the coefficient on the x to the zero term , or it 's the constant term . now , given that you have a general quadratic equation like this , the quadratic formula tells us that the solutions to this equation are x is equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . and i know it seems crazy and convoluted and hard for you to memorize right now , but as you get a lot more practice you 'll see that it actually is a pretty reasonable formula to stick in your brain someplace . and you might say , gee , this is a wacky formula , where did it come from ? and in the next video i 'm going to show you where it came from . but i want you to get used to using it first . but it really just came from completing the square on this equation right there . if you complete the square here , you 're actually going to get this solution and that is the quadratic formula , right there . so let 's apply it to some problems . let 's start off with something that we could have factored just to verify that it 's giving us the same answer . so let 's say we have x squared plus 4x minus 21 is equal to 0 . so in this situation -- let me do that in a different color -- a is equal to 1 , right ? the coefficient on the x squared term is 1. b is equal to 4 , the coefficient on the x-term . and then c is equal to negative 21 , the constant term . and let 's just plug it in the formula , so what do we get ? we get x , this tells us that x is going to be equal to negative b . negative b is negative 4 -- i put the negative sign in front of that -- negative b plus or minus the square root of b squared . b squared is 16 , right ? 4 squared is 16 , minus 4 times a , which is 1 , times c , which is negative 21 . so we can put a 21 out there and that negative sign will cancel out just like that with that -- since this is the first time we 're doing it , let me not skip too many steps . so negative 21 , just so you can see how it fit in , and then all of that over 2a . a is 1 , so all of that over 2 . so what does this simplify , or hopefully it simplifies ? so we get x is equal to negative 4 plus or minus the square root of -- let 's see we have a negative times a negative , that 's going to give us a positive . and we had 16 plus , let 's see this is 6 , 4 times 1 is 4 times 21 is 84 . 16 plus 84 is 100 . that 's nice . that 's a nice perfect square . all of that over 2 , and so this is going to be equal to negative 4 plus or minus 10 over 2 . we could just divide both of these terms by 2 right now . so this is equal to negative 4 divided by 2 is negative 2 plus or minus 10 divided by 2 is 5 . so that tells us that x could be equal to negative 2 plus 5 , which is 3 , or x could be equal to negative 2 minus 5 , which is negative 7 . so the quadratic formula seems to have given us an answer for this . you can verify just by substituting back in that these do work , or you could even just try to factor this right here . you say what two numbers when you take their product , you get negative 21 and when you take their sum you get positive 4 ? so you 'd get x plus 7 times x minus 3 is equal to negative 21 . notice 7 times negative 3 is negative 21 , 7 minus 3 is positive 4 . you would get x plus -- sorry it 's not negative -- 21 is equal to 0 . there should be a 0 there . so you get x plus 7 is equal to 0 , or x minus 3 is equal to 0 . x could be equal to negative 7 or x could be equal to 3 . so it definitely gives us the same answer as factoring , so you might say , hey why bother with this crazy mess ? and the reason we want to bother with this crazy mess is it 'll also work for problems that are hard to factor . and let 's do a couple of those , let 's do some hard-to-factor problems right now . so let 's scroll down to get some fresh real estate . let 's rewrite the formula again , just in case we have n't had it memorized yet . x is going to be equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . i 'll supply this to another problem . let 's say we have the equation 3x squared plus 6x is equal to negative 10 . well , the first thing we want to do is get it in the form where all of our terms or on the left-hand side , so let 's add 10 to both sides of this equation . we get 3x squared plus the 6x plus 10 is equal to 0 . and now we can use a quadratic formula . so let 's apply it here . so a is equal to 3 . that is a , this is b and this right here is c. so the quadratic formula tells us the solutions to this equation . the roots of this quadratic function , i guess we could call it . x is going to be equal to negative b. b is 6 , so negative 6 plus or minus the square root of b squared . b is 6 , so we get 6 squared minus 4 times a , which is 3 times c , which is 10 . let 's stretch out the radical little bit , all of that over 2 times a , 2 times 3 . so we get x is equal to negative 6 plus or minus the square root of 36 minus -- this is interesting -- minus 4 times 3 times 10 . so this is minus -- 4 times 3 times 10 . so this is minus 120 . all of that over 6 . so this is interesting , you might already realize why it 's interesting . what is this going to simplify to ? 36 minus 120 is what ? that 's 84 . we make this into a 10 , this will become an 11 , this is a 4 . it is 84 , so this is going to be equal to negative 6 plus or minus the square root of -- but not positive 84 , that 's if it 's 120 minus 36 . we have 36 minus 120 . it 's going to be negative 84 all of that 6 . so you might say , gee , this is crazy . what a this silly quadratic formula you 're introducing me to , sal ? it 's worthless . it just gives me a square root of a negative number . it 's not giving me an answer . and the reason why it 's not giving you an answer , at least an answer that you might want , is because this will have no real solutions . in the future , we 're going to introduce something called an imaginary number , which is a square root of a negative number , and then we can actually express this in terms of those numbers . so this actually does have solutions , but they involve imaginary numbers . so this actually has no real solutions , we 're taking the square root of a negative number . so the b squared with the b squared minus 4ac , if this term right here is negative , then you 're not going to have any real solutions . and let 's verify that for ourselves . let 's get our graphic calculator out and let 's graph this equation right here . so , let 's get the graphs that y is equal to -- that 's what i had there before -- 3x squared plus 6x plus 10 . so that 's the equation and we 're going to see where it intersects the x-axis . where does it equal 0 ? so let me graph it . notice , this thing just comes down and then goes back up . its vertex is sitting here above the x-axis and it 's upward-opening . it never intersects the x-axis . so at no point will this expression , will this function , equal 0 . at no point will y equal 0 on this graph . so once again , the quadratic formula seems to be working . let 's do one more example , you can never see enough examples here . and i want to do ones that are , you know , maybe not so obvious to factor . so let 's say we get negative 3x squared plus 12x plus 1 is equal to 0 . now let 's try to do it just having the quadratic formula in our brain . so the x 's that satisfy this equation are going to be negative b . this is b so negative b is negative 12 plus or minus the square root of b squared , of 144 , that 's b squared minus 4 times a , which is negative 3 times c , which is 1 , all of that over 2 times a , over 2 times negative 3 . so all of that over negative 6 , this is going to be equal to negative 12 plus or minus the square root of -- what is this ? it 's a negative times a negative so they cancel out . so i have 144 plus 12 , so that is 156 , right ? 144 plus 12 , all of that over negative 6 . now , i suspect we can simplify this 156 . we could maybe bring some things out of the radical sign . so let 's attempt to do that . so let 's do a prime factorization of 156 . sometimes , this is the hardest part , simplifying the radical . so 156 is the same thing as 2 times 78 . 78 is the same thing as 2 times what ? that 's 2 times 39 . so the square root of 156 is equal to the square root of 2 times 2 times 39 or we could say that 's the square root of 2 times 2 times the square root of 39 . and this , obviously , is just going to be the square root of 4 or this is the square root of 2 times 2 is just 2 . 2 square roots of 39 , if i did that properly , let 's see , 4 times 39 . yeah , it looks like it 's right . so this up here will simplify to negative 12 plus or minus 2 times the square root of 39 , all of that over negative 6 . now we can divide the numerator and the denominator maybe by 2 . so this will be equal to negative 6 plus or minus the square root of 39 over negative 3 . or we could separate these two terms out . we could say this is equal to negative 6 over negative 3 plus or minus the square root of 39 over negative 3 . now , this is just a 2 right here , right ? these cancel out , 6 divided by 3 is 2 , so we get 2 . and now notice , if this is plus and we use this minus sign , the plus will become negative and the negative will become positive . but it still does n't matter , right ? we could say minus or plus , that 's the same thing as plus or minus the square root of 39 nine over 3 . i think that 's about as simple as we can get this answered . i want to make a very clear point of what i did that last step . i did not forget about this negative sign . i just said it does n't matter . it 's going to turn the positive into the negative ; it 's going to turn the negative into the positive . let me rewrite this . so this right here can be rewritten as 2 plus the square root of 39 over negative 3 or 2 minus the square root of 39 over negative 3 , right ? that 's what the plus or minus means , it could be this or that or both of them , really . now in this situation , this negative 3 will turn into 2 minus the square root of 39 over 3 , right ? i 'm just taking this negative out . here the negative and the negative will become a positive , and you get 2 plus the square root of 39 over 3 , right ? a negative times a negative is a positive . so once again , you have 2 plus or minus the square of 39 over 3 . 2 plus or minus the square root of 39 over 3 are solutions to this equation right there . let verify . i 'm just curious what the graph looks like . so let 's just look at it . let me clear this . where is the clear button ? so we have negative 3 three squared plus 12x plus 1 and let 's graph it . let 's see where it intersects the x-axis . it goes up there and then back down again . so 2 plus or minus the square , you see -- the square root of 39 is going to be a little bit more than 6 , right ? because 36 is 6 squared . so it 's going be a little bit more than 6 , so this is going to be a little bit more than 2 . a little bit more than 6 divided by 2 is a little bit more than 2 . so you 're going to get one value that 's a little bit more than 4 and then another value that should be a little bit less than 1 . and that looks like the case , you have 1 , 2 , 3 , 4 . you have a value that 's pretty close to 4 , and then you have another value that is a little bit -- it looks close to 0 but maybe a little bit less than that . so anyway , hopefully you found this application of the quadratic formula helpful .
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we get 3x squared plus the 6x plus 10 is equal to 0 . and now we can use a quadratic formula . so let 's apply it here .
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is the quadratic formula generally learned in algebra 1 or 2 ?
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in this video , i 'm going to expose you to what is maybe one of at least the top five most useful formulas in mathematics . and if you 've seen many of my videos , you know that i 'm not a big fan of memorizing things . but i will recommend you memorize it with the caveat that you also remember how to prove it , because i do n't want you to just remember things and not know where they came from . but with that said , let me show you what i 'm talking about : it 's the quadratic formula . and as you might guess , it is to solve for the roots , or the zeroes of quadratic equations . so let 's speak in very general terms and i 'll show you some examples . so let 's say i have an equation of the form ax squared plus bx plus c is equal to 0 . you should recognize this . this is a quadratic equation where a , b and c are -- well , a is the coefficient on the x squared term or the second degree term , b is the coefficient on the x term and then c , is , you could imagine , the coefficient on the x to the zero term , or it 's the constant term . now , given that you have a general quadratic equation like this , the quadratic formula tells us that the solutions to this equation are x is equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . and i know it seems crazy and convoluted and hard for you to memorize right now , but as you get a lot more practice you 'll see that it actually is a pretty reasonable formula to stick in your brain someplace . and you might say , gee , this is a wacky formula , where did it come from ? and in the next video i 'm going to show you where it came from . but i want you to get used to using it first . but it really just came from completing the square on this equation right there . if you complete the square here , you 're actually going to get this solution and that is the quadratic formula , right there . so let 's apply it to some problems . let 's start off with something that we could have factored just to verify that it 's giving us the same answer . so let 's say we have x squared plus 4x minus 21 is equal to 0 . so in this situation -- let me do that in a different color -- a is equal to 1 , right ? the coefficient on the x squared term is 1. b is equal to 4 , the coefficient on the x-term . and then c is equal to negative 21 , the constant term . and let 's just plug it in the formula , so what do we get ? we get x , this tells us that x is going to be equal to negative b . negative b is negative 4 -- i put the negative sign in front of that -- negative b plus or minus the square root of b squared . b squared is 16 , right ? 4 squared is 16 , minus 4 times a , which is 1 , times c , which is negative 21 . so we can put a 21 out there and that negative sign will cancel out just like that with that -- since this is the first time we 're doing it , let me not skip too many steps . so negative 21 , just so you can see how it fit in , and then all of that over 2a . a is 1 , so all of that over 2 . so what does this simplify , or hopefully it simplifies ? so we get x is equal to negative 4 plus or minus the square root of -- let 's see we have a negative times a negative , that 's going to give us a positive . and we had 16 plus , let 's see this is 6 , 4 times 1 is 4 times 21 is 84 . 16 plus 84 is 100 . that 's nice . that 's a nice perfect square . all of that over 2 , and so this is going to be equal to negative 4 plus or minus 10 over 2 . we could just divide both of these terms by 2 right now . so this is equal to negative 4 divided by 2 is negative 2 plus or minus 10 divided by 2 is 5 . so that tells us that x could be equal to negative 2 plus 5 , which is 3 , or x could be equal to negative 2 minus 5 , which is negative 7 . so the quadratic formula seems to have given us an answer for this . you can verify just by substituting back in that these do work , or you could even just try to factor this right here . you say what two numbers when you take their product , you get negative 21 and when you take their sum you get positive 4 ? so you 'd get x plus 7 times x minus 3 is equal to negative 21 . notice 7 times negative 3 is negative 21 , 7 minus 3 is positive 4 . you would get x plus -- sorry it 's not negative -- 21 is equal to 0 . there should be a 0 there . so you get x plus 7 is equal to 0 , or x minus 3 is equal to 0 . x could be equal to negative 7 or x could be equal to 3 . so it definitely gives us the same answer as factoring , so you might say , hey why bother with this crazy mess ? and the reason we want to bother with this crazy mess is it 'll also work for problems that are hard to factor . and let 's do a couple of those , let 's do some hard-to-factor problems right now . so let 's scroll down to get some fresh real estate . let 's rewrite the formula again , just in case we have n't had it memorized yet . x is going to be equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . i 'll supply this to another problem . let 's say we have the equation 3x squared plus 6x is equal to negative 10 . well , the first thing we want to do is get it in the form where all of our terms or on the left-hand side , so let 's add 10 to both sides of this equation . we get 3x squared plus the 6x plus 10 is equal to 0 . and now we can use a quadratic formula . so let 's apply it here . so a is equal to 3 . that is a , this is b and this right here is c. so the quadratic formula tells us the solutions to this equation . the roots of this quadratic function , i guess we could call it . x is going to be equal to negative b. b is 6 , so negative 6 plus or minus the square root of b squared . b is 6 , so we get 6 squared minus 4 times a , which is 3 times c , which is 10 . let 's stretch out the radical little bit , all of that over 2 times a , 2 times 3 . so we get x is equal to negative 6 plus or minus the square root of 36 minus -- this is interesting -- minus 4 times 3 times 10 . so this is minus -- 4 times 3 times 10 . so this is minus 120 . all of that over 6 . so this is interesting , you might already realize why it 's interesting . what is this going to simplify to ? 36 minus 120 is what ? that 's 84 . we make this into a 10 , this will become an 11 , this is a 4 . it is 84 , so this is going to be equal to negative 6 plus or minus the square root of -- but not positive 84 , that 's if it 's 120 minus 36 . we have 36 minus 120 . it 's going to be negative 84 all of that 6 . so you might say , gee , this is crazy . what a this silly quadratic formula you 're introducing me to , sal ? it 's worthless . it just gives me a square root of a negative number . it 's not giving me an answer . and the reason why it 's not giving you an answer , at least an answer that you might want , is because this will have no real solutions . in the future , we 're going to introduce something called an imaginary number , which is a square root of a negative number , and then we can actually express this in terms of those numbers . so this actually does have solutions , but they involve imaginary numbers . so this actually has no real solutions , we 're taking the square root of a negative number . so the b squared with the b squared minus 4ac , if this term right here is negative , then you 're not going to have any real solutions . and let 's verify that for ourselves . let 's get our graphic calculator out and let 's graph this equation right here . so , let 's get the graphs that y is equal to -- that 's what i had there before -- 3x squared plus 6x plus 10 . so that 's the equation and we 're going to see where it intersects the x-axis . where does it equal 0 ? so let me graph it . notice , this thing just comes down and then goes back up . its vertex is sitting here above the x-axis and it 's upward-opening . it never intersects the x-axis . so at no point will this expression , will this function , equal 0 . at no point will y equal 0 on this graph . so once again , the quadratic formula seems to be working . let 's do one more example , you can never see enough examples here . and i want to do ones that are , you know , maybe not so obvious to factor . so let 's say we get negative 3x squared plus 12x plus 1 is equal to 0 . now let 's try to do it just having the quadratic formula in our brain . so the x 's that satisfy this equation are going to be negative b . this is b so negative b is negative 12 plus or minus the square root of b squared , of 144 , that 's b squared minus 4 times a , which is negative 3 times c , which is 1 , all of that over 2 times a , over 2 times negative 3 . so all of that over negative 6 , this is going to be equal to negative 12 plus or minus the square root of -- what is this ? it 's a negative times a negative so they cancel out . so i have 144 plus 12 , so that is 156 , right ? 144 plus 12 , all of that over negative 6 . now , i suspect we can simplify this 156 . we could maybe bring some things out of the radical sign . so let 's attempt to do that . so let 's do a prime factorization of 156 . sometimes , this is the hardest part , simplifying the radical . so 156 is the same thing as 2 times 78 . 78 is the same thing as 2 times what ? that 's 2 times 39 . so the square root of 156 is equal to the square root of 2 times 2 times 39 or we could say that 's the square root of 2 times 2 times the square root of 39 . and this , obviously , is just going to be the square root of 4 or this is the square root of 2 times 2 is just 2 . 2 square roots of 39 , if i did that properly , let 's see , 4 times 39 . yeah , it looks like it 's right . so this up here will simplify to negative 12 plus or minus 2 times the square root of 39 , all of that over negative 6 . now we can divide the numerator and the denominator maybe by 2 . so this will be equal to negative 6 plus or minus the square root of 39 over negative 3 . or we could separate these two terms out . we could say this is equal to negative 6 over negative 3 plus or minus the square root of 39 over negative 3 . now , this is just a 2 right here , right ? these cancel out , 6 divided by 3 is 2 , so we get 2 . and now notice , if this is plus and we use this minus sign , the plus will become negative and the negative will become positive . but it still does n't matter , right ? we could say minus or plus , that 's the same thing as plus or minus the square root of 39 nine over 3 . i think that 's about as simple as we can get this answered . i want to make a very clear point of what i did that last step . i did not forget about this negative sign . i just said it does n't matter . it 's going to turn the positive into the negative ; it 's going to turn the negative into the positive . let me rewrite this . so this right here can be rewritten as 2 plus the square root of 39 over negative 3 or 2 minus the square root of 39 over negative 3 , right ? that 's what the plus or minus means , it could be this or that or both of them , really . now in this situation , this negative 3 will turn into 2 minus the square root of 39 over 3 , right ? i 'm just taking this negative out . here the negative and the negative will become a positive , and you get 2 plus the square root of 39 over 3 , right ? a negative times a negative is a positive . so once again , you have 2 plus or minus the square of 39 over 3 . 2 plus or minus the square root of 39 over 3 are solutions to this equation right there . let verify . i 'm just curious what the graph looks like . so let 's just look at it . let me clear this . where is the clear button ? so we have negative 3 three squared plus 12x plus 1 and let 's graph it . let 's see where it intersects the x-axis . it goes up there and then back down again . so 2 plus or minus the square , you see -- the square root of 39 is going to be a little bit more than 6 , right ? because 36 is 6 squared . so it 's going be a little bit more than 6 , so this is going to be a little bit more than 2 . a little bit more than 6 divided by 2 is a little bit more than 2 . so you 're going to get one value that 's a little bit more than 4 and then another value that should be a little bit less than 1 . and that looks like the case , you have 1 , 2 , 3 , 4 . you have a value that 's pretty close to 4 , and then you have another value that is a little bit -- it looks close to 0 but maybe a little bit less than that . so anyway , hopefully you found this application of the quadratic formula helpful .
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you would get x plus -- sorry it 's not negative -- 21 is equal to 0 . there should be a 0 there . so you get x plus 7 is equal to 0 , or x minus 3 is equal to 0 . x could be equal to negative 7 or x could be equal to 3 .
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is the quadratic equation used to find the age of the universe , like 13.798 +/- 0.037 x 10^9 years , the +/- sign means there is two answers ?
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in this video , i 'm going to expose you to what is maybe one of at least the top five most useful formulas in mathematics . and if you 've seen many of my videos , you know that i 'm not a big fan of memorizing things . but i will recommend you memorize it with the caveat that you also remember how to prove it , because i do n't want you to just remember things and not know where they came from . but with that said , let me show you what i 'm talking about : it 's the quadratic formula . and as you might guess , it is to solve for the roots , or the zeroes of quadratic equations . so let 's speak in very general terms and i 'll show you some examples . so let 's say i have an equation of the form ax squared plus bx plus c is equal to 0 . you should recognize this . this is a quadratic equation where a , b and c are -- well , a is the coefficient on the x squared term or the second degree term , b is the coefficient on the x term and then c , is , you could imagine , the coefficient on the x to the zero term , or it 's the constant term . now , given that you have a general quadratic equation like this , the quadratic formula tells us that the solutions to this equation are x is equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . and i know it seems crazy and convoluted and hard for you to memorize right now , but as you get a lot more practice you 'll see that it actually is a pretty reasonable formula to stick in your brain someplace . and you might say , gee , this is a wacky formula , where did it come from ? and in the next video i 'm going to show you where it came from . but i want you to get used to using it first . but it really just came from completing the square on this equation right there . if you complete the square here , you 're actually going to get this solution and that is the quadratic formula , right there . so let 's apply it to some problems . let 's start off with something that we could have factored just to verify that it 's giving us the same answer . so let 's say we have x squared plus 4x minus 21 is equal to 0 . so in this situation -- let me do that in a different color -- a is equal to 1 , right ? the coefficient on the x squared term is 1. b is equal to 4 , the coefficient on the x-term . and then c is equal to negative 21 , the constant term . and let 's just plug it in the formula , so what do we get ? we get x , this tells us that x is going to be equal to negative b . negative b is negative 4 -- i put the negative sign in front of that -- negative b plus or minus the square root of b squared . b squared is 16 , right ? 4 squared is 16 , minus 4 times a , which is 1 , times c , which is negative 21 . so we can put a 21 out there and that negative sign will cancel out just like that with that -- since this is the first time we 're doing it , let me not skip too many steps . so negative 21 , just so you can see how it fit in , and then all of that over 2a . a is 1 , so all of that over 2 . so what does this simplify , or hopefully it simplifies ? so we get x is equal to negative 4 plus or minus the square root of -- let 's see we have a negative times a negative , that 's going to give us a positive . and we had 16 plus , let 's see this is 6 , 4 times 1 is 4 times 21 is 84 . 16 plus 84 is 100 . that 's nice . that 's a nice perfect square . all of that over 2 , and so this is going to be equal to negative 4 plus or minus 10 over 2 . we could just divide both of these terms by 2 right now . so this is equal to negative 4 divided by 2 is negative 2 plus or minus 10 divided by 2 is 5 . so that tells us that x could be equal to negative 2 plus 5 , which is 3 , or x could be equal to negative 2 minus 5 , which is negative 7 . so the quadratic formula seems to have given us an answer for this . you can verify just by substituting back in that these do work , or you could even just try to factor this right here . you say what two numbers when you take their product , you get negative 21 and when you take their sum you get positive 4 ? so you 'd get x plus 7 times x minus 3 is equal to negative 21 . notice 7 times negative 3 is negative 21 , 7 minus 3 is positive 4 . you would get x plus -- sorry it 's not negative -- 21 is equal to 0 . there should be a 0 there . so you get x plus 7 is equal to 0 , or x minus 3 is equal to 0 . x could be equal to negative 7 or x could be equal to 3 . so it definitely gives us the same answer as factoring , so you might say , hey why bother with this crazy mess ? and the reason we want to bother with this crazy mess is it 'll also work for problems that are hard to factor . and let 's do a couple of those , let 's do some hard-to-factor problems right now . so let 's scroll down to get some fresh real estate . let 's rewrite the formula again , just in case we have n't had it memorized yet . x is going to be equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . i 'll supply this to another problem . let 's say we have the equation 3x squared plus 6x is equal to negative 10 . well , the first thing we want to do is get it in the form where all of our terms or on the left-hand side , so let 's add 10 to both sides of this equation . we get 3x squared plus the 6x plus 10 is equal to 0 . and now we can use a quadratic formula . so let 's apply it here . so a is equal to 3 . that is a , this is b and this right here is c. so the quadratic formula tells us the solutions to this equation . the roots of this quadratic function , i guess we could call it . x is going to be equal to negative b. b is 6 , so negative 6 plus or minus the square root of b squared . b is 6 , so we get 6 squared minus 4 times a , which is 3 times c , which is 10 . let 's stretch out the radical little bit , all of that over 2 times a , 2 times 3 . so we get x is equal to negative 6 plus or minus the square root of 36 minus -- this is interesting -- minus 4 times 3 times 10 . so this is minus -- 4 times 3 times 10 . so this is minus 120 . all of that over 6 . so this is interesting , you might already realize why it 's interesting . what is this going to simplify to ? 36 minus 120 is what ? that 's 84 . we make this into a 10 , this will become an 11 , this is a 4 . it is 84 , so this is going to be equal to negative 6 plus or minus the square root of -- but not positive 84 , that 's if it 's 120 minus 36 . we have 36 minus 120 . it 's going to be negative 84 all of that 6 . so you might say , gee , this is crazy . what a this silly quadratic formula you 're introducing me to , sal ? it 's worthless . it just gives me a square root of a negative number . it 's not giving me an answer . and the reason why it 's not giving you an answer , at least an answer that you might want , is because this will have no real solutions . in the future , we 're going to introduce something called an imaginary number , which is a square root of a negative number , and then we can actually express this in terms of those numbers . so this actually does have solutions , but they involve imaginary numbers . so this actually has no real solutions , we 're taking the square root of a negative number . so the b squared with the b squared minus 4ac , if this term right here is negative , then you 're not going to have any real solutions . and let 's verify that for ourselves . let 's get our graphic calculator out and let 's graph this equation right here . so , let 's get the graphs that y is equal to -- that 's what i had there before -- 3x squared plus 6x plus 10 . so that 's the equation and we 're going to see where it intersects the x-axis . where does it equal 0 ? so let me graph it . notice , this thing just comes down and then goes back up . its vertex is sitting here above the x-axis and it 's upward-opening . it never intersects the x-axis . so at no point will this expression , will this function , equal 0 . at no point will y equal 0 on this graph . so once again , the quadratic formula seems to be working . let 's do one more example , you can never see enough examples here . and i want to do ones that are , you know , maybe not so obvious to factor . so let 's say we get negative 3x squared plus 12x plus 1 is equal to 0 . now let 's try to do it just having the quadratic formula in our brain . so the x 's that satisfy this equation are going to be negative b . this is b so negative b is negative 12 plus or minus the square root of b squared , of 144 , that 's b squared minus 4 times a , which is negative 3 times c , which is 1 , all of that over 2 times a , over 2 times negative 3 . so all of that over negative 6 , this is going to be equal to negative 12 plus or minus the square root of -- what is this ? it 's a negative times a negative so they cancel out . so i have 144 plus 12 , so that is 156 , right ? 144 plus 12 , all of that over negative 6 . now , i suspect we can simplify this 156 . we could maybe bring some things out of the radical sign . so let 's attempt to do that . so let 's do a prime factorization of 156 . sometimes , this is the hardest part , simplifying the radical . so 156 is the same thing as 2 times 78 . 78 is the same thing as 2 times what ? that 's 2 times 39 . so the square root of 156 is equal to the square root of 2 times 2 times 39 or we could say that 's the square root of 2 times 2 times the square root of 39 . and this , obviously , is just going to be the square root of 4 or this is the square root of 2 times 2 is just 2 . 2 square roots of 39 , if i did that properly , let 's see , 4 times 39 . yeah , it looks like it 's right . so this up here will simplify to negative 12 plus or minus 2 times the square root of 39 , all of that over negative 6 . now we can divide the numerator and the denominator maybe by 2 . so this will be equal to negative 6 plus or minus the square root of 39 over negative 3 . or we could separate these two terms out . we could say this is equal to negative 6 over negative 3 plus or minus the square root of 39 over negative 3 . now , this is just a 2 right here , right ? these cancel out , 6 divided by 3 is 2 , so we get 2 . and now notice , if this is plus and we use this minus sign , the plus will become negative and the negative will become positive . but it still does n't matter , right ? we could say minus or plus , that 's the same thing as plus or minus the square root of 39 nine over 3 . i think that 's about as simple as we can get this answered . i want to make a very clear point of what i did that last step . i did not forget about this negative sign . i just said it does n't matter . it 's going to turn the positive into the negative ; it 's going to turn the negative into the positive . let me rewrite this . so this right here can be rewritten as 2 plus the square root of 39 over negative 3 or 2 minus the square root of 39 over negative 3 , right ? that 's what the plus or minus means , it could be this or that or both of them , really . now in this situation , this negative 3 will turn into 2 minus the square root of 39 over 3 , right ? i 'm just taking this negative out . here the negative and the negative will become a positive , and you get 2 plus the square root of 39 over 3 , right ? a negative times a negative is a positive . so once again , you have 2 plus or minus the square of 39 over 3 . 2 plus or minus the square root of 39 over 3 are solutions to this equation right there . let verify . i 'm just curious what the graph looks like . so let 's just look at it . let me clear this . where is the clear button ? so we have negative 3 three squared plus 12x plus 1 and let 's graph it . let 's see where it intersects the x-axis . it goes up there and then back down again . so 2 plus or minus the square , you see -- the square root of 39 is going to be a little bit more than 6 , right ? because 36 is 6 squared . so it 's going be a little bit more than 6 , so this is going to be a little bit more than 2 . a little bit more than 6 divided by 2 is a little bit more than 2 . so you 're going to get one value that 's a little bit more than 4 and then another value that should be a little bit less than 1 . and that looks like the case , you have 1 , 2 , 3 , 4 . you have a value that 's pretty close to 4 , and then you have another value that is a little bit -- it looks close to 0 but maybe a little bit less than that . so anyway , hopefully you found this application of the quadratic formula helpful .
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and let 's just plug it in the formula , so what do we get ? we get x , this tells us that x is going to be equal to negative b . negative b is negative 4 -- i put the negative sign in front of that -- negative b plus or minus the square root of b squared .
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for example , the equation 20x^2+196x+288 i plugged the correct corresponding numbers into the quadratic formula and got two numbers but how would i put them into the ( x+a ) ( x+b ) groupings when the 20 is in front of the x ?
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in this video , i 'm going to expose you to what is maybe one of at least the top five most useful formulas in mathematics . and if you 've seen many of my videos , you know that i 'm not a big fan of memorizing things . but i will recommend you memorize it with the caveat that you also remember how to prove it , because i do n't want you to just remember things and not know where they came from . but with that said , let me show you what i 'm talking about : it 's the quadratic formula . and as you might guess , it is to solve for the roots , or the zeroes of quadratic equations . so let 's speak in very general terms and i 'll show you some examples . so let 's say i have an equation of the form ax squared plus bx plus c is equal to 0 . you should recognize this . this is a quadratic equation where a , b and c are -- well , a is the coefficient on the x squared term or the second degree term , b is the coefficient on the x term and then c , is , you could imagine , the coefficient on the x to the zero term , or it 's the constant term . now , given that you have a general quadratic equation like this , the quadratic formula tells us that the solutions to this equation are x is equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . and i know it seems crazy and convoluted and hard for you to memorize right now , but as you get a lot more practice you 'll see that it actually is a pretty reasonable formula to stick in your brain someplace . and you might say , gee , this is a wacky formula , where did it come from ? and in the next video i 'm going to show you where it came from . but i want you to get used to using it first . but it really just came from completing the square on this equation right there . if you complete the square here , you 're actually going to get this solution and that is the quadratic formula , right there . so let 's apply it to some problems . let 's start off with something that we could have factored just to verify that it 's giving us the same answer . so let 's say we have x squared plus 4x minus 21 is equal to 0 . so in this situation -- let me do that in a different color -- a is equal to 1 , right ? the coefficient on the x squared term is 1. b is equal to 4 , the coefficient on the x-term . and then c is equal to negative 21 , the constant term . and let 's just plug it in the formula , so what do we get ? we get x , this tells us that x is going to be equal to negative b . negative b is negative 4 -- i put the negative sign in front of that -- negative b plus or minus the square root of b squared . b squared is 16 , right ? 4 squared is 16 , minus 4 times a , which is 1 , times c , which is negative 21 . so we can put a 21 out there and that negative sign will cancel out just like that with that -- since this is the first time we 're doing it , let me not skip too many steps . so negative 21 , just so you can see how it fit in , and then all of that over 2a . a is 1 , so all of that over 2 . so what does this simplify , or hopefully it simplifies ? so we get x is equal to negative 4 plus or minus the square root of -- let 's see we have a negative times a negative , that 's going to give us a positive . and we had 16 plus , let 's see this is 6 , 4 times 1 is 4 times 21 is 84 . 16 plus 84 is 100 . that 's nice . that 's a nice perfect square . all of that over 2 , and so this is going to be equal to negative 4 plus or minus 10 over 2 . we could just divide both of these terms by 2 right now . so this is equal to negative 4 divided by 2 is negative 2 plus or minus 10 divided by 2 is 5 . so that tells us that x could be equal to negative 2 plus 5 , which is 3 , or x could be equal to negative 2 minus 5 , which is negative 7 . so the quadratic formula seems to have given us an answer for this . you can verify just by substituting back in that these do work , or you could even just try to factor this right here . you say what two numbers when you take their product , you get negative 21 and when you take their sum you get positive 4 ? so you 'd get x plus 7 times x minus 3 is equal to negative 21 . notice 7 times negative 3 is negative 21 , 7 minus 3 is positive 4 . you would get x plus -- sorry it 's not negative -- 21 is equal to 0 . there should be a 0 there . so you get x plus 7 is equal to 0 , or x minus 3 is equal to 0 . x could be equal to negative 7 or x could be equal to 3 . so it definitely gives us the same answer as factoring , so you might say , hey why bother with this crazy mess ? and the reason we want to bother with this crazy mess is it 'll also work for problems that are hard to factor . and let 's do a couple of those , let 's do some hard-to-factor problems right now . so let 's scroll down to get some fresh real estate . let 's rewrite the formula again , just in case we have n't had it memorized yet . x is going to be equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . i 'll supply this to another problem . let 's say we have the equation 3x squared plus 6x is equal to negative 10 . well , the first thing we want to do is get it in the form where all of our terms or on the left-hand side , so let 's add 10 to both sides of this equation . we get 3x squared plus the 6x plus 10 is equal to 0 . and now we can use a quadratic formula . so let 's apply it here . so a is equal to 3 . that is a , this is b and this right here is c. so the quadratic formula tells us the solutions to this equation . the roots of this quadratic function , i guess we could call it . x is going to be equal to negative b. b is 6 , so negative 6 plus or minus the square root of b squared . b is 6 , so we get 6 squared minus 4 times a , which is 3 times c , which is 10 . let 's stretch out the radical little bit , all of that over 2 times a , 2 times 3 . so we get x is equal to negative 6 plus or minus the square root of 36 minus -- this is interesting -- minus 4 times 3 times 10 . so this is minus -- 4 times 3 times 10 . so this is minus 120 . all of that over 6 . so this is interesting , you might already realize why it 's interesting . what is this going to simplify to ? 36 minus 120 is what ? that 's 84 . we make this into a 10 , this will become an 11 , this is a 4 . it is 84 , so this is going to be equal to negative 6 plus or minus the square root of -- but not positive 84 , that 's if it 's 120 minus 36 . we have 36 minus 120 . it 's going to be negative 84 all of that 6 . so you might say , gee , this is crazy . what a this silly quadratic formula you 're introducing me to , sal ? it 's worthless . it just gives me a square root of a negative number . it 's not giving me an answer . and the reason why it 's not giving you an answer , at least an answer that you might want , is because this will have no real solutions . in the future , we 're going to introduce something called an imaginary number , which is a square root of a negative number , and then we can actually express this in terms of those numbers . so this actually does have solutions , but they involve imaginary numbers . so this actually has no real solutions , we 're taking the square root of a negative number . so the b squared with the b squared minus 4ac , if this term right here is negative , then you 're not going to have any real solutions . and let 's verify that for ourselves . let 's get our graphic calculator out and let 's graph this equation right here . so , let 's get the graphs that y is equal to -- that 's what i had there before -- 3x squared plus 6x plus 10 . so that 's the equation and we 're going to see where it intersects the x-axis . where does it equal 0 ? so let me graph it . notice , this thing just comes down and then goes back up . its vertex is sitting here above the x-axis and it 's upward-opening . it never intersects the x-axis . so at no point will this expression , will this function , equal 0 . at no point will y equal 0 on this graph . so once again , the quadratic formula seems to be working . let 's do one more example , you can never see enough examples here . and i want to do ones that are , you know , maybe not so obvious to factor . so let 's say we get negative 3x squared plus 12x plus 1 is equal to 0 . now let 's try to do it just having the quadratic formula in our brain . so the x 's that satisfy this equation are going to be negative b . this is b so negative b is negative 12 plus or minus the square root of b squared , of 144 , that 's b squared minus 4 times a , which is negative 3 times c , which is 1 , all of that over 2 times a , over 2 times negative 3 . so all of that over negative 6 , this is going to be equal to negative 12 plus or minus the square root of -- what is this ? it 's a negative times a negative so they cancel out . so i have 144 plus 12 , so that is 156 , right ? 144 plus 12 , all of that over negative 6 . now , i suspect we can simplify this 156 . we could maybe bring some things out of the radical sign . so let 's attempt to do that . so let 's do a prime factorization of 156 . sometimes , this is the hardest part , simplifying the radical . so 156 is the same thing as 2 times 78 . 78 is the same thing as 2 times what ? that 's 2 times 39 . so the square root of 156 is equal to the square root of 2 times 2 times 39 or we could say that 's the square root of 2 times 2 times the square root of 39 . and this , obviously , is just going to be the square root of 4 or this is the square root of 2 times 2 is just 2 . 2 square roots of 39 , if i did that properly , let 's see , 4 times 39 . yeah , it looks like it 's right . so this up here will simplify to negative 12 plus or minus 2 times the square root of 39 , all of that over negative 6 . now we can divide the numerator and the denominator maybe by 2 . so this will be equal to negative 6 plus or minus the square root of 39 over negative 3 . or we could separate these two terms out . we could say this is equal to negative 6 over negative 3 plus or minus the square root of 39 over negative 3 . now , this is just a 2 right here , right ? these cancel out , 6 divided by 3 is 2 , so we get 2 . and now notice , if this is plus and we use this minus sign , the plus will become negative and the negative will become positive . but it still does n't matter , right ? we could say minus or plus , that 's the same thing as plus or minus the square root of 39 nine over 3 . i think that 's about as simple as we can get this answered . i want to make a very clear point of what i did that last step . i did not forget about this negative sign . i just said it does n't matter . it 's going to turn the positive into the negative ; it 's going to turn the negative into the positive . let me rewrite this . so this right here can be rewritten as 2 plus the square root of 39 over negative 3 or 2 minus the square root of 39 over negative 3 , right ? that 's what the plus or minus means , it could be this or that or both of them , really . now in this situation , this negative 3 will turn into 2 minus the square root of 39 over 3 , right ? i 'm just taking this negative out . here the negative and the negative will become a positive , and you get 2 plus the square root of 39 over 3 , right ? a negative times a negative is a positive . so once again , you have 2 plus or minus the square of 39 over 3 . 2 plus or minus the square root of 39 over 3 are solutions to this equation right there . let verify . i 'm just curious what the graph looks like . so let 's just look at it . let me clear this . where is the clear button ? so we have negative 3 three squared plus 12x plus 1 and let 's graph it . let 's see where it intersects the x-axis . it goes up there and then back down again . so 2 plus or minus the square , you see -- the square root of 39 is going to be a little bit more than 6 , right ? because 36 is 6 squared . so it 's going be a little bit more than 6 , so this is going to be a little bit more than 2 . a little bit more than 6 divided by 2 is a little bit more than 2 . so you 're going to get one value that 's a little bit more than 4 and then another value that should be a little bit less than 1 . and that looks like the case , you have 1 , 2 , 3 , 4 . you have a value that 's pretty close to 4 , and then you have another value that is a little bit -- it looks close to 0 but maybe a little bit less than that . so anyway , hopefully you found this application of the quadratic formula helpful .
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its vertex is sitting here above the x-axis and it 's upward-opening . it never intersects the x-axis . so at no point will this expression , will this function , equal 0 .
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could we say then that the purpose of the quadratic equation is to find the intersections at the x-axis and that if it has `` no real solutions '' it simply means that the graph of the equation is somewhere above the x-axis ?
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in this video , i 'm going to expose you to what is maybe one of at least the top five most useful formulas in mathematics . and if you 've seen many of my videos , you know that i 'm not a big fan of memorizing things . but i will recommend you memorize it with the caveat that you also remember how to prove it , because i do n't want you to just remember things and not know where they came from . but with that said , let me show you what i 'm talking about : it 's the quadratic formula . and as you might guess , it is to solve for the roots , or the zeroes of quadratic equations . so let 's speak in very general terms and i 'll show you some examples . so let 's say i have an equation of the form ax squared plus bx plus c is equal to 0 . you should recognize this . this is a quadratic equation where a , b and c are -- well , a is the coefficient on the x squared term or the second degree term , b is the coefficient on the x term and then c , is , you could imagine , the coefficient on the x to the zero term , or it 's the constant term . now , given that you have a general quadratic equation like this , the quadratic formula tells us that the solutions to this equation are x is equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . and i know it seems crazy and convoluted and hard for you to memorize right now , but as you get a lot more practice you 'll see that it actually is a pretty reasonable formula to stick in your brain someplace . and you might say , gee , this is a wacky formula , where did it come from ? and in the next video i 'm going to show you where it came from . but i want you to get used to using it first . but it really just came from completing the square on this equation right there . if you complete the square here , you 're actually going to get this solution and that is the quadratic formula , right there . so let 's apply it to some problems . let 's start off with something that we could have factored just to verify that it 's giving us the same answer . so let 's say we have x squared plus 4x minus 21 is equal to 0 . so in this situation -- let me do that in a different color -- a is equal to 1 , right ? the coefficient on the x squared term is 1. b is equal to 4 , the coefficient on the x-term . and then c is equal to negative 21 , the constant term . and let 's just plug it in the formula , so what do we get ? we get x , this tells us that x is going to be equal to negative b . negative b is negative 4 -- i put the negative sign in front of that -- negative b plus or minus the square root of b squared . b squared is 16 , right ? 4 squared is 16 , minus 4 times a , which is 1 , times c , which is negative 21 . so we can put a 21 out there and that negative sign will cancel out just like that with that -- since this is the first time we 're doing it , let me not skip too many steps . so negative 21 , just so you can see how it fit in , and then all of that over 2a . a is 1 , so all of that over 2 . so what does this simplify , or hopefully it simplifies ? so we get x is equal to negative 4 plus or minus the square root of -- let 's see we have a negative times a negative , that 's going to give us a positive . and we had 16 plus , let 's see this is 6 , 4 times 1 is 4 times 21 is 84 . 16 plus 84 is 100 . that 's nice . that 's a nice perfect square . all of that over 2 , and so this is going to be equal to negative 4 plus or minus 10 over 2 . we could just divide both of these terms by 2 right now . so this is equal to negative 4 divided by 2 is negative 2 plus or minus 10 divided by 2 is 5 . so that tells us that x could be equal to negative 2 plus 5 , which is 3 , or x could be equal to negative 2 minus 5 , which is negative 7 . so the quadratic formula seems to have given us an answer for this . you can verify just by substituting back in that these do work , or you could even just try to factor this right here . you say what two numbers when you take their product , you get negative 21 and when you take their sum you get positive 4 ? so you 'd get x plus 7 times x minus 3 is equal to negative 21 . notice 7 times negative 3 is negative 21 , 7 minus 3 is positive 4 . you would get x plus -- sorry it 's not negative -- 21 is equal to 0 . there should be a 0 there . so you get x plus 7 is equal to 0 , or x minus 3 is equal to 0 . x could be equal to negative 7 or x could be equal to 3 . so it definitely gives us the same answer as factoring , so you might say , hey why bother with this crazy mess ? and the reason we want to bother with this crazy mess is it 'll also work for problems that are hard to factor . and let 's do a couple of those , let 's do some hard-to-factor problems right now . so let 's scroll down to get some fresh real estate . let 's rewrite the formula again , just in case we have n't had it memorized yet . x is going to be equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . i 'll supply this to another problem . let 's say we have the equation 3x squared plus 6x is equal to negative 10 . well , the first thing we want to do is get it in the form where all of our terms or on the left-hand side , so let 's add 10 to both sides of this equation . we get 3x squared plus the 6x plus 10 is equal to 0 . and now we can use a quadratic formula . so let 's apply it here . so a is equal to 3 . that is a , this is b and this right here is c. so the quadratic formula tells us the solutions to this equation . the roots of this quadratic function , i guess we could call it . x is going to be equal to negative b. b is 6 , so negative 6 plus or minus the square root of b squared . b is 6 , so we get 6 squared minus 4 times a , which is 3 times c , which is 10 . let 's stretch out the radical little bit , all of that over 2 times a , 2 times 3 . so we get x is equal to negative 6 plus or minus the square root of 36 minus -- this is interesting -- minus 4 times 3 times 10 . so this is minus -- 4 times 3 times 10 . so this is minus 120 . all of that over 6 . so this is interesting , you might already realize why it 's interesting . what is this going to simplify to ? 36 minus 120 is what ? that 's 84 . we make this into a 10 , this will become an 11 , this is a 4 . it is 84 , so this is going to be equal to negative 6 plus or minus the square root of -- but not positive 84 , that 's if it 's 120 minus 36 . we have 36 minus 120 . it 's going to be negative 84 all of that 6 . so you might say , gee , this is crazy . what a this silly quadratic formula you 're introducing me to , sal ? it 's worthless . it just gives me a square root of a negative number . it 's not giving me an answer . and the reason why it 's not giving you an answer , at least an answer that you might want , is because this will have no real solutions . in the future , we 're going to introduce something called an imaginary number , which is a square root of a negative number , and then we can actually express this in terms of those numbers . so this actually does have solutions , but they involve imaginary numbers . so this actually has no real solutions , we 're taking the square root of a negative number . so the b squared with the b squared minus 4ac , if this term right here is negative , then you 're not going to have any real solutions . and let 's verify that for ourselves . let 's get our graphic calculator out and let 's graph this equation right here . so , let 's get the graphs that y is equal to -- that 's what i had there before -- 3x squared plus 6x plus 10 . so that 's the equation and we 're going to see where it intersects the x-axis . where does it equal 0 ? so let me graph it . notice , this thing just comes down and then goes back up . its vertex is sitting here above the x-axis and it 's upward-opening . it never intersects the x-axis . so at no point will this expression , will this function , equal 0 . at no point will y equal 0 on this graph . so once again , the quadratic formula seems to be working . let 's do one more example , you can never see enough examples here . and i want to do ones that are , you know , maybe not so obvious to factor . so let 's say we get negative 3x squared plus 12x plus 1 is equal to 0 . now let 's try to do it just having the quadratic formula in our brain . so the x 's that satisfy this equation are going to be negative b . this is b so negative b is negative 12 plus or minus the square root of b squared , of 144 , that 's b squared minus 4 times a , which is negative 3 times c , which is 1 , all of that over 2 times a , over 2 times negative 3 . so all of that over negative 6 , this is going to be equal to negative 12 plus or minus the square root of -- what is this ? it 's a negative times a negative so they cancel out . so i have 144 plus 12 , so that is 156 , right ? 144 plus 12 , all of that over negative 6 . now , i suspect we can simplify this 156 . we could maybe bring some things out of the radical sign . so let 's attempt to do that . so let 's do a prime factorization of 156 . sometimes , this is the hardest part , simplifying the radical . so 156 is the same thing as 2 times 78 . 78 is the same thing as 2 times what ? that 's 2 times 39 . so the square root of 156 is equal to the square root of 2 times 2 times 39 or we could say that 's the square root of 2 times 2 times the square root of 39 . and this , obviously , is just going to be the square root of 4 or this is the square root of 2 times 2 is just 2 . 2 square roots of 39 , if i did that properly , let 's see , 4 times 39 . yeah , it looks like it 's right . so this up here will simplify to negative 12 plus or minus 2 times the square root of 39 , all of that over negative 6 . now we can divide the numerator and the denominator maybe by 2 . so this will be equal to negative 6 plus or minus the square root of 39 over negative 3 . or we could separate these two terms out . we could say this is equal to negative 6 over negative 3 plus or minus the square root of 39 over negative 3 . now , this is just a 2 right here , right ? these cancel out , 6 divided by 3 is 2 , so we get 2 . and now notice , if this is plus and we use this minus sign , the plus will become negative and the negative will become positive . but it still does n't matter , right ? we could say minus or plus , that 's the same thing as plus or minus the square root of 39 nine over 3 . i think that 's about as simple as we can get this answered . i want to make a very clear point of what i did that last step . i did not forget about this negative sign . i just said it does n't matter . it 's going to turn the positive into the negative ; it 's going to turn the negative into the positive . let me rewrite this . so this right here can be rewritten as 2 plus the square root of 39 over negative 3 or 2 minus the square root of 39 over negative 3 , right ? that 's what the plus or minus means , it could be this or that or both of them , really . now in this situation , this negative 3 will turn into 2 minus the square root of 39 over 3 , right ? i 'm just taking this negative out . here the negative and the negative will become a positive , and you get 2 plus the square root of 39 over 3 , right ? a negative times a negative is a positive . so once again , you have 2 plus or minus the square of 39 over 3 . 2 plus or minus the square root of 39 over 3 are solutions to this equation right there . let verify . i 'm just curious what the graph looks like . so let 's just look at it . let me clear this . where is the clear button ? so we have negative 3 three squared plus 12x plus 1 and let 's graph it . let 's see where it intersects the x-axis . it goes up there and then back down again . so 2 plus or minus the square , you see -- the square root of 39 is going to be a little bit more than 6 , right ? because 36 is 6 squared . so it 's going be a little bit more than 6 , so this is going to be a little bit more than 2 . a little bit more than 6 divided by 2 is a little bit more than 2 . so you 're going to get one value that 's a little bit more than 4 and then another value that should be a little bit less than 1 . and that looks like the case , you have 1 , 2 , 3 , 4 . you have a value that 's pretty close to 4 , and then you have another value that is a little bit -- it looks close to 0 but maybe a little bit less than that . so anyway , hopefully you found this application of the quadratic formula helpful .
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it 's worthless . it just gives me a square root of a negative number . it 's not giving me an answer .
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in a quadratic formula , how would you factor a square root of a negative number ?
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in this video , i 'm going to expose you to what is maybe one of at least the top five most useful formulas in mathematics . and if you 've seen many of my videos , you know that i 'm not a big fan of memorizing things . but i will recommend you memorize it with the caveat that you also remember how to prove it , because i do n't want you to just remember things and not know where they came from . but with that said , let me show you what i 'm talking about : it 's the quadratic formula . and as you might guess , it is to solve for the roots , or the zeroes of quadratic equations . so let 's speak in very general terms and i 'll show you some examples . so let 's say i have an equation of the form ax squared plus bx plus c is equal to 0 . you should recognize this . this is a quadratic equation where a , b and c are -- well , a is the coefficient on the x squared term or the second degree term , b is the coefficient on the x term and then c , is , you could imagine , the coefficient on the x to the zero term , or it 's the constant term . now , given that you have a general quadratic equation like this , the quadratic formula tells us that the solutions to this equation are x is equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . and i know it seems crazy and convoluted and hard for you to memorize right now , but as you get a lot more practice you 'll see that it actually is a pretty reasonable formula to stick in your brain someplace . and you might say , gee , this is a wacky formula , where did it come from ? and in the next video i 'm going to show you where it came from . but i want you to get used to using it first . but it really just came from completing the square on this equation right there . if you complete the square here , you 're actually going to get this solution and that is the quadratic formula , right there . so let 's apply it to some problems . let 's start off with something that we could have factored just to verify that it 's giving us the same answer . so let 's say we have x squared plus 4x minus 21 is equal to 0 . so in this situation -- let me do that in a different color -- a is equal to 1 , right ? the coefficient on the x squared term is 1. b is equal to 4 , the coefficient on the x-term . and then c is equal to negative 21 , the constant term . and let 's just plug it in the formula , so what do we get ? we get x , this tells us that x is going to be equal to negative b . negative b is negative 4 -- i put the negative sign in front of that -- negative b plus or minus the square root of b squared . b squared is 16 , right ? 4 squared is 16 , minus 4 times a , which is 1 , times c , which is negative 21 . so we can put a 21 out there and that negative sign will cancel out just like that with that -- since this is the first time we 're doing it , let me not skip too many steps . so negative 21 , just so you can see how it fit in , and then all of that over 2a . a is 1 , so all of that over 2 . so what does this simplify , or hopefully it simplifies ? so we get x is equal to negative 4 plus or minus the square root of -- let 's see we have a negative times a negative , that 's going to give us a positive . and we had 16 plus , let 's see this is 6 , 4 times 1 is 4 times 21 is 84 . 16 plus 84 is 100 . that 's nice . that 's a nice perfect square . all of that over 2 , and so this is going to be equal to negative 4 plus or minus 10 over 2 . we could just divide both of these terms by 2 right now . so this is equal to negative 4 divided by 2 is negative 2 plus or minus 10 divided by 2 is 5 . so that tells us that x could be equal to negative 2 plus 5 , which is 3 , or x could be equal to negative 2 minus 5 , which is negative 7 . so the quadratic formula seems to have given us an answer for this . you can verify just by substituting back in that these do work , or you could even just try to factor this right here . you say what two numbers when you take their product , you get negative 21 and when you take their sum you get positive 4 ? so you 'd get x plus 7 times x minus 3 is equal to negative 21 . notice 7 times negative 3 is negative 21 , 7 minus 3 is positive 4 . you would get x plus -- sorry it 's not negative -- 21 is equal to 0 . there should be a 0 there . so you get x plus 7 is equal to 0 , or x minus 3 is equal to 0 . x could be equal to negative 7 or x could be equal to 3 . so it definitely gives us the same answer as factoring , so you might say , hey why bother with this crazy mess ? and the reason we want to bother with this crazy mess is it 'll also work for problems that are hard to factor . and let 's do a couple of those , let 's do some hard-to-factor problems right now . so let 's scroll down to get some fresh real estate . let 's rewrite the formula again , just in case we have n't had it memorized yet . x is going to be equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . i 'll supply this to another problem . let 's say we have the equation 3x squared plus 6x is equal to negative 10 . well , the first thing we want to do is get it in the form where all of our terms or on the left-hand side , so let 's add 10 to both sides of this equation . we get 3x squared plus the 6x plus 10 is equal to 0 . and now we can use a quadratic formula . so let 's apply it here . so a is equal to 3 . that is a , this is b and this right here is c. so the quadratic formula tells us the solutions to this equation . the roots of this quadratic function , i guess we could call it . x is going to be equal to negative b. b is 6 , so negative 6 plus or minus the square root of b squared . b is 6 , so we get 6 squared minus 4 times a , which is 3 times c , which is 10 . let 's stretch out the radical little bit , all of that over 2 times a , 2 times 3 . so we get x is equal to negative 6 plus or minus the square root of 36 minus -- this is interesting -- minus 4 times 3 times 10 . so this is minus -- 4 times 3 times 10 . so this is minus 120 . all of that over 6 . so this is interesting , you might already realize why it 's interesting . what is this going to simplify to ? 36 minus 120 is what ? that 's 84 . we make this into a 10 , this will become an 11 , this is a 4 . it is 84 , so this is going to be equal to negative 6 plus or minus the square root of -- but not positive 84 , that 's if it 's 120 minus 36 . we have 36 minus 120 . it 's going to be negative 84 all of that 6 . so you might say , gee , this is crazy . what a this silly quadratic formula you 're introducing me to , sal ? it 's worthless . it just gives me a square root of a negative number . it 's not giving me an answer . and the reason why it 's not giving you an answer , at least an answer that you might want , is because this will have no real solutions . in the future , we 're going to introduce something called an imaginary number , which is a square root of a negative number , and then we can actually express this in terms of those numbers . so this actually does have solutions , but they involve imaginary numbers . so this actually has no real solutions , we 're taking the square root of a negative number . so the b squared with the b squared minus 4ac , if this term right here is negative , then you 're not going to have any real solutions . and let 's verify that for ourselves . let 's get our graphic calculator out and let 's graph this equation right here . so , let 's get the graphs that y is equal to -- that 's what i had there before -- 3x squared plus 6x plus 10 . so that 's the equation and we 're going to see where it intersects the x-axis . where does it equal 0 ? so let me graph it . notice , this thing just comes down and then goes back up . its vertex is sitting here above the x-axis and it 's upward-opening . it never intersects the x-axis . so at no point will this expression , will this function , equal 0 . at no point will y equal 0 on this graph . so once again , the quadratic formula seems to be working . let 's do one more example , you can never see enough examples here . and i want to do ones that are , you know , maybe not so obvious to factor . so let 's say we get negative 3x squared plus 12x plus 1 is equal to 0 . now let 's try to do it just having the quadratic formula in our brain . so the x 's that satisfy this equation are going to be negative b . this is b so negative b is negative 12 plus or minus the square root of b squared , of 144 , that 's b squared minus 4 times a , which is negative 3 times c , which is 1 , all of that over 2 times a , over 2 times negative 3 . so all of that over negative 6 , this is going to be equal to negative 12 plus or minus the square root of -- what is this ? it 's a negative times a negative so they cancel out . so i have 144 plus 12 , so that is 156 , right ? 144 plus 12 , all of that over negative 6 . now , i suspect we can simplify this 156 . we could maybe bring some things out of the radical sign . so let 's attempt to do that . so let 's do a prime factorization of 156 . sometimes , this is the hardest part , simplifying the radical . so 156 is the same thing as 2 times 78 . 78 is the same thing as 2 times what ? that 's 2 times 39 . so the square root of 156 is equal to the square root of 2 times 2 times 39 or we could say that 's the square root of 2 times 2 times the square root of 39 . and this , obviously , is just going to be the square root of 4 or this is the square root of 2 times 2 is just 2 . 2 square roots of 39 , if i did that properly , let 's see , 4 times 39 . yeah , it looks like it 's right . so this up here will simplify to negative 12 plus or minus 2 times the square root of 39 , all of that over negative 6 . now we can divide the numerator and the denominator maybe by 2 . so this will be equal to negative 6 plus or minus the square root of 39 over negative 3 . or we could separate these two terms out . we could say this is equal to negative 6 over negative 3 plus or minus the square root of 39 over negative 3 . now , this is just a 2 right here , right ? these cancel out , 6 divided by 3 is 2 , so we get 2 . and now notice , if this is plus and we use this minus sign , the plus will become negative and the negative will become positive . but it still does n't matter , right ? we could say minus or plus , that 's the same thing as plus or minus the square root of 39 nine over 3 . i think that 's about as simple as we can get this answered . i want to make a very clear point of what i did that last step . i did not forget about this negative sign . i just said it does n't matter . it 's going to turn the positive into the negative ; it 's going to turn the negative into the positive . let me rewrite this . so this right here can be rewritten as 2 plus the square root of 39 over negative 3 or 2 minus the square root of 39 over negative 3 , right ? that 's what the plus or minus means , it could be this or that or both of them , really . now in this situation , this negative 3 will turn into 2 minus the square root of 39 over 3 , right ? i 'm just taking this negative out . here the negative and the negative will become a positive , and you get 2 plus the square root of 39 over 3 , right ? a negative times a negative is a positive . so once again , you have 2 plus or minus the square of 39 over 3 . 2 plus or minus the square root of 39 over 3 are solutions to this equation right there . let verify . i 'm just curious what the graph looks like . so let 's just look at it . let me clear this . where is the clear button ? so we have negative 3 three squared plus 12x plus 1 and let 's graph it . let 's see where it intersects the x-axis . it goes up there and then back down again . so 2 plus or minus the square , you see -- the square root of 39 is going to be a little bit more than 6 , right ? because 36 is 6 squared . so it 's going be a little bit more than 6 , so this is going to be a little bit more than 2 . a little bit more than 6 divided by 2 is a little bit more than 2 . so you 're going to get one value that 's a little bit more than 4 and then another value that should be a little bit less than 1 . and that looks like the case , you have 1 , 2 , 3 , 4 . you have a value that 's pretty close to 4 , and then you have another value that is a little bit -- it looks close to 0 but maybe a little bit less than that . so anyway , hopefully you found this application of the quadratic formula helpful .
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that 's 2 times 39 . so the square root of 156 is equal to the square root of 2 times 2 times 39 or we could say that 's the square root of 2 times 2 times the square root of 39 . and this , obviously , is just going to be the square root of 4 or this is the square root of 2 times 2 is just 2 .
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at 1 why does sal write the square root of 156 as the square root of 2 * 2 times the square root of 39 ?
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in this video , i 'm going to expose you to what is maybe one of at least the top five most useful formulas in mathematics . and if you 've seen many of my videos , you know that i 'm not a big fan of memorizing things . but i will recommend you memorize it with the caveat that you also remember how to prove it , because i do n't want you to just remember things and not know where they came from . but with that said , let me show you what i 'm talking about : it 's the quadratic formula . and as you might guess , it is to solve for the roots , or the zeroes of quadratic equations . so let 's speak in very general terms and i 'll show you some examples . so let 's say i have an equation of the form ax squared plus bx plus c is equal to 0 . you should recognize this . this is a quadratic equation where a , b and c are -- well , a is the coefficient on the x squared term or the second degree term , b is the coefficient on the x term and then c , is , you could imagine , the coefficient on the x to the zero term , or it 's the constant term . now , given that you have a general quadratic equation like this , the quadratic formula tells us that the solutions to this equation are x is equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . and i know it seems crazy and convoluted and hard for you to memorize right now , but as you get a lot more practice you 'll see that it actually is a pretty reasonable formula to stick in your brain someplace . and you might say , gee , this is a wacky formula , where did it come from ? and in the next video i 'm going to show you where it came from . but i want you to get used to using it first . but it really just came from completing the square on this equation right there . if you complete the square here , you 're actually going to get this solution and that is the quadratic formula , right there . so let 's apply it to some problems . let 's start off with something that we could have factored just to verify that it 's giving us the same answer . so let 's say we have x squared plus 4x minus 21 is equal to 0 . so in this situation -- let me do that in a different color -- a is equal to 1 , right ? the coefficient on the x squared term is 1. b is equal to 4 , the coefficient on the x-term . and then c is equal to negative 21 , the constant term . and let 's just plug it in the formula , so what do we get ? we get x , this tells us that x is going to be equal to negative b . negative b is negative 4 -- i put the negative sign in front of that -- negative b plus or minus the square root of b squared . b squared is 16 , right ? 4 squared is 16 , minus 4 times a , which is 1 , times c , which is negative 21 . so we can put a 21 out there and that negative sign will cancel out just like that with that -- since this is the first time we 're doing it , let me not skip too many steps . so negative 21 , just so you can see how it fit in , and then all of that over 2a . a is 1 , so all of that over 2 . so what does this simplify , or hopefully it simplifies ? so we get x is equal to negative 4 plus or minus the square root of -- let 's see we have a negative times a negative , that 's going to give us a positive . and we had 16 plus , let 's see this is 6 , 4 times 1 is 4 times 21 is 84 . 16 plus 84 is 100 . that 's nice . that 's a nice perfect square . all of that over 2 , and so this is going to be equal to negative 4 plus or minus 10 over 2 . we could just divide both of these terms by 2 right now . so this is equal to negative 4 divided by 2 is negative 2 plus or minus 10 divided by 2 is 5 . so that tells us that x could be equal to negative 2 plus 5 , which is 3 , or x could be equal to negative 2 minus 5 , which is negative 7 . so the quadratic formula seems to have given us an answer for this . you can verify just by substituting back in that these do work , or you could even just try to factor this right here . you say what two numbers when you take their product , you get negative 21 and when you take their sum you get positive 4 ? so you 'd get x plus 7 times x minus 3 is equal to negative 21 . notice 7 times negative 3 is negative 21 , 7 minus 3 is positive 4 . you would get x plus -- sorry it 's not negative -- 21 is equal to 0 . there should be a 0 there . so you get x plus 7 is equal to 0 , or x minus 3 is equal to 0 . x could be equal to negative 7 or x could be equal to 3 . so it definitely gives us the same answer as factoring , so you might say , hey why bother with this crazy mess ? and the reason we want to bother with this crazy mess is it 'll also work for problems that are hard to factor . and let 's do a couple of those , let 's do some hard-to-factor problems right now . so let 's scroll down to get some fresh real estate . let 's rewrite the formula again , just in case we have n't had it memorized yet . x is going to be equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . i 'll supply this to another problem . let 's say we have the equation 3x squared plus 6x is equal to negative 10 . well , the first thing we want to do is get it in the form where all of our terms or on the left-hand side , so let 's add 10 to both sides of this equation . we get 3x squared plus the 6x plus 10 is equal to 0 . and now we can use a quadratic formula . so let 's apply it here . so a is equal to 3 . that is a , this is b and this right here is c. so the quadratic formula tells us the solutions to this equation . the roots of this quadratic function , i guess we could call it . x is going to be equal to negative b. b is 6 , so negative 6 plus or minus the square root of b squared . b is 6 , so we get 6 squared minus 4 times a , which is 3 times c , which is 10 . let 's stretch out the radical little bit , all of that over 2 times a , 2 times 3 . so we get x is equal to negative 6 plus or minus the square root of 36 minus -- this is interesting -- minus 4 times 3 times 10 . so this is minus -- 4 times 3 times 10 . so this is minus 120 . all of that over 6 . so this is interesting , you might already realize why it 's interesting . what is this going to simplify to ? 36 minus 120 is what ? that 's 84 . we make this into a 10 , this will become an 11 , this is a 4 . it is 84 , so this is going to be equal to negative 6 plus or minus the square root of -- but not positive 84 , that 's if it 's 120 minus 36 . we have 36 minus 120 . it 's going to be negative 84 all of that 6 . so you might say , gee , this is crazy . what a this silly quadratic formula you 're introducing me to , sal ? it 's worthless . it just gives me a square root of a negative number . it 's not giving me an answer . and the reason why it 's not giving you an answer , at least an answer that you might want , is because this will have no real solutions . in the future , we 're going to introduce something called an imaginary number , which is a square root of a negative number , and then we can actually express this in terms of those numbers . so this actually does have solutions , but they involve imaginary numbers . so this actually has no real solutions , we 're taking the square root of a negative number . so the b squared with the b squared minus 4ac , if this term right here is negative , then you 're not going to have any real solutions . and let 's verify that for ourselves . let 's get our graphic calculator out and let 's graph this equation right here . so , let 's get the graphs that y is equal to -- that 's what i had there before -- 3x squared plus 6x plus 10 . so that 's the equation and we 're going to see where it intersects the x-axis . where does it equal 0 ? so let me graph it . notice , this thing just comes down and then goes back up . its vertex is sitting here above the x-axis and it 's upward-opening . it never intersects the x-axis . so at no point will this expression , will this function , equal 0 . at no point will y equal 0 on this graph . so once again , the quadratic formula seems to be working . let 's do one more example , you can never see enough examples here . and i want to do ones that are , you know , maybe not so obvious to factor . so let 's say we get negative 3x squared plus 12x plus 1 is equal to 0 . now let 's try to do it just having the quadratic formula in our brain . so the x 's that satisfy this equation are going to be negative b . this is b so negative b is negative 12 plus or minus the square root of b squared , of 144 , that 's b squared minus 4 times a , which is negative 3 times c , which is 1 , all of that over 2 times a , over 2 times negative 3 . so all of that over negative 6 , this is going to be equal to negative 12 plus or minus the square root of -- what is this ? it 's a negative times a negative so they cancel out . so i have 144 plus 12 , so that is 156 , right ? 144 plus 12 , all of that over negative 6 . now , i suspect we can simplify this 156 . we could maybe bring some things out of the radical sign . so let 's attempt to do that . so let 's do a prime factorization of 156 . sometimes , this is the hardest part , simplifying the radical . so 156 is the same thing as 2 times 78 . 78 is the same thing as 2 times what ? that 's 2 times 39 . so the square root of 156 is equal to the square root of 2 times 2 times 39 or we could say that 's the square root of 2 times 2 times the square root of 39 . and this , obviously , is just going to be the square root of 4 or this is the square root of 2 times 2 is just 2 . 2 square roots of 39 , if i did that properly , let 's see , 4 times 39 . yeah , it looks like it 's right . so this up here will simplify to negative 12 plus or minus 2 times the square root of 39 , all of that over negative 6 . now we can divide the numerator and the denominator maybe by 2 . so this will be equal to negative 6 plus or minus the square root of 39 over negative 3 . or we could separate these two terms out . we could say this is equal to negative 6 over negative 3 plus or minus the square root of 39 over negative 3 . now , this is just a 2 right here , right ? these cancel out , 6 divided by 3 is 2 , so we get 2 . and now notice , if this is plus and we use this minus sign , the plus will become negative and the negative will become positive . but it still does n't matter , right ? we could say minus or plus , that 's the same thing as plus or minus the square root of 39 nine over 3 . i think that 's about as simple as we can get this answered . i want to make a very clear point of what i did that last step . i did not forget about this negative sign . i just said it does n't matter . it 's going to turn the positive into the negative ; it 's going to turn the negative into the positive . let me rewrite this . so this right here can be rewritten as 2 plus the square root of 39 over negative 3 or 2 minus the square root of 39 over negative 3 , right ? that 's what the plus or minus means , it could be this or that or both of them , really . now in this situation , this negative 3 will turn into 2 minus the square root of 39 over 3 , right ? i 'm just taking this negative out . here the negative and the negative will become a positive , and you get 2 plus the square root of 39 over 3 , right ? a negative times a negative is a positive . so once again , you have 2 plus or minus the square of 39 over 3 . 2 plus or minus the square root of 39 over 3 are solutions to this equation right there . let verify . i 'm just curious what the graph looks like . so let 's just look at it . let me clear this . where is the clear button ? so we have negative 3 three squared plus 12x plus 1 and let 's graph it . let 's see where it intersects the x-axis . it goes up there and then back down again . so 2 plus or minus the square , you see -- the square root of 39 is going to be a little bit more than 6 , right ? because 36 is 6 squared . so it 's going be a little bit more than 6 , so this is going to be a little bit more than 2 . a little bit more than 6 divided by 2 is a little bit more than 2 . so you 're going to get one value that 's a little bit more than 4 and then another value that should be a little bit less than 1 . and that looks like the case , you have 1 , 2 , 3 , 4 . you have a value that 's pretty close to 4 , and then you have another value that is a little bit -- it looks close to 0 but maybe a little bit less than that . so anyway , hopefully you found this application of the quadratic formula helpful .
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in the future , we 're going to introduce something called an imaginary number , which is a square root of a negative number , and then we can actually express this in terms of those numbers . so this actually does have solutions , but they involve imaginary numbers . so this actually has no real solutions , we 're taking the square root of a negative number .
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how are real numbers different from complex numbers ?
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in this video , i 'm going to expose you to what is maybe one of at least the top five most useful formulas in mathematics . and if you 've seen many of my videos , you know that i 'm not a big fan of memorizing things . but i will recommend you memorize it with the caveat that you also remember how to prove it , because i do n't want you to just remember things and not know where they came from . but with that said , let me show you what i 'm talking about : it 's the quadratic formula . and as you might guess , it is to solve for the roots , or the zeroes of quadratic equations . so let 's speak in very general terms and i 'll show you some examples . so let 's say i have an equation of the form ax squared plus bx plus c is equal to 0 . you should recognize this . this is a quadratic equation where a , b and c are -- well , a is the coefficient on the x squared term or the second degree term , b is the coefficient on the x term and then c , is , you could imagine , the coefficient on the x to the zero term , or it 's the constant term . now , given that you have a general quadratic equation like this , the quadratic formula tells us that the solutions to this equation are x is equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . and i know it seems crazy and convoluted and hard for you to memorize right now , but as you get a lot more practice you 'll see that it actually is a pretty reasonable formula to stick in your brain someplace . and you might say , gee , this is a wacky formula , where did it come from ? and in the next video i 'm going to show you where it came from . but i want you to get used to using it first . but it really just came from completing the square on this equation right there . if you complete the square here , you 're actually going to get this solution and that is the quadratic formula , right there . so let 's apply it to some problems . let 's start off with something that we could have factored just to verify that it 's giving us the same answer . so let 's say we have x squared plus 4x minus 21 is equal to 0 . so in this situation -- let me do that in a different color -- a is equal to 1 , right ? the coefficient on the x squared term is 1. b is equal to 4 , the coefficient on the x-term . and then c is equal to negative 21 , the constant term . and let 's just plug it in the formula , so what do we get ? we get x , this tells us that x is going to be equal to negative b . negative b is negative 4 -- i put the negative sign in front of that -- negative b plus or minus the square root of b squared . b squared is 16 , right ? 4 squared is 16 , minus 4 times a , which is 1 , times c , which is negative 21 . so we can put a 21 out there and that negative sign will cancel out just like that with that -- since this is the first time we 're doing it , let me not skip too many steps . so negative 21 , just so you can see how it fit in , and then all of that over 2a . a is 1 , so all of that over 2 . so what does this simplify , or hopefully it simplifies ? so we get x is equal to negative 4 plus or minus the square root of -- let 's see we have a negative times a negative , that 's going to give us a positive . and we had 16 plus , let 's see this is 6 , 4 times 1 is 4 times 21 is 84 . 16 plus 84 is 100 . that 's nice . that 's a nice perfect square . all of that over 2 , and so this is going to be equal to negative 4 plus or minus 10 over 2 . we could just divide both of these terms by 2 right now . so this is equal to negative 4 divided by 2 is negative 2 plus or minus 10 divided by 2 is 5 . so that tells us that x could be equal to negative 2 plus 5 , which is 3 , or x could be equal to negative 2 minus 5 , which is negative 7 . so the quadratic formula seems to have given us an answer for this . you can verify just by substituting back in that these do work , or you could even just try to factor this right here . you say what two numbers when you take their product , you get negative 21 and when you take their sum you get positive 4 ? so you 'd get x plus 7 times x minus 3 is equal to negative 21 . notice 7 times negative 3 is negative 21 , 7 minus 3 is positive 4 . you would get x plus -- sorry it 's not negative -- 21 is equal to 0 . there should be a 0 there . so you get x plus 7 is equal to 0 , or x minus 3 is equal to 0 . x could be equal to negative 7 or x could be equal to 3 . so it definitely gives us the same answer as factoring , so you might say , hey why bother with this crazy mess ? and the reason we want to bother with this crazy mess is it 'll also work for problems that are hard to factor . and let 's do a couple of those , let 's do some hard-to-factor problems right now . so let 's scroll down to get some fresh real estate . let 's rewrite the formula again , just in case we have n't had it memorized yet . x is going to be equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . i 'll supply this to another problem . let 's say we have the equation 3x squared plus 6x is equal to negative 10 . well , the first thing we want to do is get it in the form where all of our terms or on the left-hand side , so let 's add 10 to both sides of this equation . we get 3x squared plus the 6x plus 10 is equal to 0 . and now we can use a quadratic formula . so let 's apply it here . so a is equal to 3 . that is a , this is b and this right here is c. so the quadratic formula tells us the solutions to this equation . the roots of this quadratic function , i guess we could call it . x is going to be equal to negative b. b is 6 , so negative 6 plus or minus the square root of b squared . b is 6 , so we get 6 squared minus 4 times a , which is 3 times c , which is 10 . let 's stretch out the radical little bit , all of that over 2 times a , 2 times 3 . so we get x is equal to negative 6 plus or minus the square root of 36 minus -- this is interesting -- minus 4 times 3 times 10 . so this is minus -- 4 times 3 times 10 . so this is minus 120 . all of that over 6 . so this is interesting , you might already realize why it 's interesting . what is this going to simplify to ? 36 minus 120 is what ? that 's 84 . we make this into a 10 , this will become an 11 , this is a 4 . it is 84 , so this is going to be equal to negative 6 plus or minus the square root of -- but not positive 84 , that 's if it 's 120 minus 36 . we have 36 minus 120 . it 's going to be negative 84 all of that 6 . so you might say , gee , this is crazy . what a this silly quadratic formula you 're introducing me to , sal ? it 's worthless . it just gives me a square root of a negative number . it 's not giving me an answer . and the reason why it 's not giving you an answer , at least an answer that you might want , is because this will have no real solutions . in the future , we 're going to introduce something called an imaginary number , which is a square root of a negative number , and then we can actually express this in terms of those numbers . so this actually does have solutions , but they involve imaginary numbers . so this actually has no real solutions , we 're taking the square root of a negative number . so the b squared with the b squared minus 4ac , if this term right here is negative , then you 're not going to have any real solutions . and let 's verify that for ourselves . let 's get our graphic calculator out and let 's graph this equation right here . so , let 's get the graphs that y is equal to -- that 's what i had there before -- 3x squared plus 6x plus 10 . so that 's the equation and we 're going to see where it intersects the x-axis . where does it equal 0 ? so let me graph it . notice , this thing just comes down and then goes back up . its vertex is sitting here above the x-axis and it 's upward-opening . it never intersects the x-axis . so at no point will this expression , will this function , equal 0 . at no point will y equal 0 on this graph . so once again , the quadratic formula seems to be working . let 's do one more example , you can never see enough examples here . and i want to do ones that are , you know , maybe not so obvious to factor . so let 's say we get negative 3x squared plus 12x plus 1 is equal to 0 . now let 's try to do it just having the quadratic formula in our brain . so the x 's that satisfy this equation are going to be negative b . this is b so negative b is negative 12 plus or minus the square root of b squared , of 144 , that 's b squared minus 4 times a , which is negative 3 times c , which is 1 , all of that over 2 times a , over 2 times negative 3 . so all of that over negative 6 , this is going to be equal to negative 12 plus or minus the square root of -- what is this ? it 's a negative times a negative so they cancel out . so i have 144 plus 12 , so that is 156 , right ? 144 plus 12 , all of that over negative 6 . now , i suspect we can simplify this 156 . we could maybe bring some things out of the radical sign . so let 's attempt to do that . so let 's do a prime factorization of 156 . sometimes , this is the hardest part , simplifying the radical . so 156 is the same thing as 2 times 78 . 78 is the same thing as 2 times what ? that 's 2 times 39 . so the square root of 156 is equal to the square root of 2 times 2 times 39 or we could say that 's the square root of 2 times 2 times the square root of 39 . and this , obviously , is just going to be the square root of 4 or this is the square root of 2 times 2 is just 2 . 2 square roots of 39 , if i did that properly , let 's see , 4 times 39 . yeah , it looks like it 's right . so this up here will simplify to negative 12 plus or minus 2 times the square root of 39 , all of that over negative 6 . now we can divide the numerator and the denominator maybe by 2 . so this will be equal to negative 6 plus or minus the square root of 39 over negative 3 . or we could separate these two terms out . we could say this is equal to negative 6 over negative 3 plus or minus the square root of 39 over negative 3 . now , this is just a 2 right here , right ? these cancel out , 6 divided by 3 is 2 , so we get 2 . and now notice , if this is plus and we use this minus sign , the plus will become negative and the negative will become positive . but it still does n't matter , right ? we could say minus or plus , that 's the same thing as plus or minus the square root of 39 nine over 3 . i think that 's about as simple as we can get this answered . i want to make a very clear point of what i did that last step . i did not forget about this negative sign . i just said it does n't matter . it 's going to turn the positive into the negative ; it 's going to turn the negative into the positive . let me rewrite this . so this right here can be rewritten as 2 plus the square root of 39 over negative 3 or 2 minus the square root of 39 over negative 3 , right ? that 's what the plus or minus means , it could be this or that or both of them , really . now in this situation , this negative 3 will turn into 2 minus the square root of 39 over 3 , right ? i 'm just taking this negative out . here the negative and the negative will become a positive , and you get 2 plus the square root of 39 over 3 , right ? a negative times a negative is a positive . so once again , you have 2 plus or minus the square of 39 over 3 . 2 plus or minus the square root of 39 over 3 are solutions to this equation right there . let verify . i 'm just curious what the graph looks like . so let 's just look at it . let me clear this . where is the clear button ? so we have negative 3 three squared plus 12x plus 1 and let 's graph it . let 's see where it intersects the x-axis . it goes up there and then back down again . so 2 plus or minus the square , you see -- the square root of 39 is going to be a little bit more than 6 , right ? because 36 is 6 squared . so it 's going be a little bit more than 6 , so this is going to be a little bit more than 2 . a little bit more than 6 divided by 2 is a little bit more than 2 . so you 're going to get one value that 's a little bit more than 4 and then another value that should be a little bit less than 1 . and that looks like the case , you have 1 , 2 , 3 , 4 . you have a value that 's pretty close to 4 , and then you have another value that is a little bit -- it looks close to 0 but maybe a little bit less than that . so anyway , hopefully you found this application of the quadratic formula helpful .
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we get 3x squared plus the 6x plus 10 is equal to 0 . and now we can use a quadratic formula . so let 's apply it here .
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so when you learn the quadratic formula knowing how to discover what x is by factoring becomes obsolete or it can be used for other purposes ?
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in this video , i 'm going to expose you to what is maybe one of at least the top five most useful formulas in mathematics . and if you 've seen many of my videos , you know that i 'm not a big fan of memorizing things . but i will recommend you memorize it with the caveat that you also remember how to prove it , because i do n't want you to just remember things and not know where they came from . but with that said , let me show you what i 'm talking about : it 's the quadratic formula . and as you might guess , it is to solve for the roots , or the zeroes of quadratic equations . so let 's speak in very general terms and i 'll show you some examples . so let 's say i have an equation of the form ax squared plus bx plus c is equal to 0 . you should recognize this . this is a quadratic equation where a , b and c are -- well , a is the coefficient on the x squared term or the second degree term , b is the coefficient on the x term and then c , is , you could imagine , the coefficient on the x to the zero term , or it 's the constant term . now , given that you have a general quadratic equation like this , the quadratic formula tells us that the solutions to this equation are x is equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . and i know it seems crazy and convoluted and hard for you to memorize right now , but as you get a lot more practice you 'll see that it actually is a pretty reasonable formula to stick in your brain someplace . and you might say , gee , this is a wacky formula , where did it come from ? and in the next video i 'm going to show you where it came from . but i want you to get used to using it first . but it really just came from completing the square on this equation right there . if you complete the square here , you 're actually going to get this solution and that is the quadratic formula , right there . so let 's apply it to some problems . let 's start off with something that we could have factored just to verify that it 's giving us the same answer . so let 's say we have x squared plus 4x minus 21 is equal to 0 . so in this situation -- let me do that in a different color -- a is equal to 1 , right ? the coefficient on the x squared term is 1. b is equal to 4 , the coefficient on the x-term . and then c is equal to negative 21 , the constant term . and let 's just plug it in the formula , so what do we get ? we get x , this tells us that x is going to be equal to negative b . negative b is negative 4 -- i put the negative sign in front of that -- negative b plus or minus the square root of b squared . b squared is 16 , right ? 4 squared is 16 , minus 4 times a , which is 1 , times c , which is negative 21 . so we can put a 21 out there and that negative sign will cancel out just like that with that -- since this is the first time we 're doing it , let me not skip too many steps . so negative 21 , just so you can see how it fit in , and then all of that over 2a . a is 1 , so all of that over 2 . so what does this simplify , or hopefully it simplifies ? so we get x is equal to negative 4 plus or minus the square root of -- let 's see we have a negative times a negative , that 's going to give us a positive . and we had 16 plus , let 's see this is 6 , 4 times 1 is 4 times 21 is 84 . 16 plus 84 is 100 . that 's nice . that 's a nice perfect square . all of that over 2 , and so this is going to be equal to negative 4 plus or minus 10 over 2 . we could just divide both of these terms by 2 right now . so this is equal to negative 4 divided by 2 is negative 2 plus or minus 10 divided by 2 is 5 . so that tells us that x could be equal to negative 2 plus 5 , which is 3 , or x could be equal to negative 2 minus 5 , which is negative 7 . so the quadratic formula seems to have given us an answer for this . you can verify just by substituting back in that these do work , or you could even just try to factor this right here . you say what two numbers when you take their product , you get negative 21 and when you take their sum you get positive 4 ? so you 'd get x plus 7 times x minus 3 is equal to negative 21 . notice 7 times negative 3 is negative 21 , 7 minus 3 is positive 4 . you would get x plus -- sorry it 's not negative -- 21 is equal to 0 . there should be a 0 there . so you get x plus 7 is equal to 0 , or x minus 3 is equal to 0 . x could be equal to negative 7 or x could be equal to 3 . so it definitely gives us the same answer as factoring , so you might say , hey why bother with this crazy mess ? and the reason we want to bother with this crazy mess is it 'll also work for problems that are hard to factor . and let 's do a couple of those , let 's do some hard-to-factor problems right now . so let 's scroll down to get some fresh real estate . let 's rewrite the formula again , just in case we have n't had it memorized yet . x is going to be equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . i 'll supply this to another problem . let 's say we have the equation 3x squared plus 6x is equal to negative 10 . well , the first thing we want to do is get it in the form where all of our terms or on the left-hand side , so let 's add 10 to both sides of this equation . we get 3x squared plus the 6x plus 10 is equal to 0 . and now we can use a quadratic formula . so let 's apply it here . so a is equal to 3 . that is a , this is b and this right here is c. so the quadratic formula tells us the solutions to this equation . the roots of this quadratic function , i guess we could call it . x is going to be equal to negative b. b is 6 , so negative 6 plus or minus the square root of b squared . b is 6 , so we get 6 squared minus 4 times a , which is 3 times c , which is 10 . let 's stretch out the radical little bit , all of that over 2 times a , 2 times 3 . so we get x is equal to negative 6 plus or minus the square root of 36 minus -- this is interesting -- minus 4 times 3 times 10 . so this is minus -- 4 times 3 times 10 . so this is minus 120 . all of that over 6 . so this is interesting , you might already realize why it 's interesting . what is this going to simplify to ? 36 minus 120 is what ? that 's 84 . we make this into a 10 , this will become an 11 , this is a 4 . it is 84 , so this is going to be equal to negative 6 plus or minus the square root of -- but not positive 84 , that 's if it 's 120 minus 36 . we have 36 minus 120 . it 's going to be negative 84 all of that 6 . so you might say , gee , this is crazy . what a this silly quadratic formula you 're introducing me to , sal ? it 's worthless . it just gives me a square root of a negative number . it 's not giving me an answer . and the reason why it 's not giving you an answer , at least an answer that you might want , is because this will have no real solutions . in the future , we 're going to introduce something called an imaginary number , which is a square root of a negative number , and then we can actually express this in terms of those numbers . so this actually does have solutions , but they involve imaginary numbers . so this actually has no real solutions , we 're taking the square root of a negative number . so the b squared with the b squared minus 4ac , if this term right here is negative , then you 're not going to have any real solutions . and let 's verify that for ourselves . let 's get our graphic calculator out and let 's graph this equation right here . so , let 's get the graphs that y is equal to -- that 's what i had there before -- 3x squared plus 6x plus 10 . so that 's the equation and we 're going to see where it intersects the x-axis . where does it equal 0 ? so let me graph it . notice , this thing just comes down and then goes back up . its vertex is sitting here above the x-axis and it 's upward-opening . it never intersects the x-axis . so at no point will this expression , will this function , equal 0 . at no point will y equal 0 on this graph . so once again , the quadratic formula seems to be working . let 's do one more example , you can never see enough examples here . and i want to do ones that are , you know , maybe not so obvious to factor . so let 's say we get negative 3x squared plus 12x plus 1 is equal to 0 . now let 's try to do it just having the quadratic formula in our brain . so the x 's that satisfy this equation are going to be negative b . this is b so negative b is negative 12 plus or minus the square root of b squared , of 144 , that 's b squared minus 4 times a , which is negative 3 times c , which is 1 , all of that over 2 times a , over 2 times negative 3 . so all of that over negative 6 , this is going to be equal to negative 12 plus or minus the square root of -- what is this ? it 's a negative times a negative so they cancel out . so i have 144 plus 12 , so that is 156 , right ? 144 plus 12 , all of that over negative 6 . now , i suspect we can simplify this 156 . we could maybe bring some things out of the radical sign . so let 's attempt to do that . so let 's do a prime factorization of 156 . sometimes , this is the hardest part , simplifying the radical . so 156 is the same thing as 2 times 78 . 78 is the same thing as 2 times what ? that 's 2 times 39 . so the square root of 156 is equal to the square root of 2 times 2 times 39 or we could say that 's the square root of 2 times 2 times the square root of 39 . and this , obviously , is just going to be the square root of 4 or this is the square root of 2 times 2 is just 2 . 2 square roots of 39 , if i did that properly , let 's see , 4 times 39 . yeah , it looks like it 's right . so this up here will simplify to negative 12 plus or minus 2 times the square root of 39 , all of that over negative 6 . now we can divide the numerator and the denominator maybe by 2 . so this will be equal to negative 6 plus or minus the square root of 39 over negative 3 . or we could separate these two terms out . we could say this is equal to negative 6 over negative 3 plus or minus the square root of 39 over negative 3 . now , this is just a 2 right here , right ? these cancel out , 6 divided by 3 is 2 , so we get 2 . and now notice , if this is plus and we use this minus sign , the plus will become negative and the negative will become positive . but it still does n't matter , right ? we could say minus or plus , that 's the same thing as plus or minus the square root of 39 nine over 3 . i think that 's about as simple as we can get this answered . i want to make a very clear point of what i did that last step . i did not forget about this negative sign . i just said it does n't matter . it 's going to turn the positive into the negative ; it 's going to turn the negative into the positive . let me rewrite this . so this right here can be rewritten as 2 plus the square root of 39 over negative 3 or 2 minus the square root of 39 over negative 3 , right ? that 's what the plus or minus means , it could be this or that or both of them , really . now in this situation , this negative 3 will turn into 2 minus the square root of 39 over 3 , right ? i 'm just taking this negative out . here the negative and the negative will become a positive , and you get 2 plus the square root of 39 over 3 , right ? a negative times a negative is a positive . so once again , you have 2 plus or minus the square of 39 over 3 . 2 plus or minus the square root of 39 over 3 are solutions to this equation right there . let verify . i 'm just curious what the graph looks like . so let 's just look at it . let me clear this . where is the clear button ? so we have negative 3 three squared plus 12x plus 1 and let 's graph it . let 's see where it intersects the x-axis . it goes up there and then back down again . so 2 plus or minus the square , you see -- the square root of 39 is going to be a little bit more than 6 , right ? because 36 is 6 squared . so it 's going be a little bit more than 6 , so this is going to be a little bit more than 2 . a little bit more than 6 divided by 2 is a little bit more than 2 . so you 're going to get one value that 's a little bit more than 4 and then another value that should be a little bit less than 1 . and that looks like the case , you have 1 , 2 , 3 , 4 . you have a value that 's pretty close to 4 , and then you have another value that is a little bit -- it looks close to 0 but maybe a little bit less than that . so anyway , hopefully you found this application of the quadratic formula helpful .
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we get 3x squared plus the 6x plus 10 is equal to 0 . and now we can use a quadratic formula . so let 's apply it here .
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when solving a quadratic using the quadratic formula is it acceptable to move move all the variables to the right side such as -9x^2=-x-4 or is it better to always move the variables to the left side ?
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in this video , i 'm going to expose you to what is maybe one of at least the top five most useful formulas in mathematics . and if you 've seen many of my videos , you know that i 'm not a big fan of memorizing things . but i will recommend you memorize it with the caveat that you also remember how to prove it , because i do n't want you to just remember things and not know where they came from . but with that said , let me show you what i 'm talking about : it 's the quadratic formula . and as you might guess , it is to solve for the roots , or the zeroes of quadratic equations . so let 's speak in very general terms and i 'll show you some examples . so let 's say i have an equation of the form ax squared plus bx plus c is equal to 0 . you should recognize this . this is a quadratic equation where a , b and c are -- well , a is the coefficient on the x squared term or the second degree term , b is the coefficient on the x term and then c , is , you could imagine , the coefficient on the x to the zero term , or it 's the constant term . now , given that you have a general quadratic equation like this , the quadratic formula tells us that the solutions to this equation are x is equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . and i know it seems crazy and convoluted and hard for you to memorize right now , but as you get a lot more practice you 'll see that it actually is a pretty reasonable formula to stick in your brain someplace . and you might say , gee , this is a wacky formula , where did it come from ? and in the next video i 'm going to show you where it came from . but i want you to get used to using it first . but it really just came from completing the square on this equation right there . if you complete the square here , you 're actually going to get this solution and that is the quadratic formula , right there . so let 's apply it to some problems . let 's start off with something that we could have factored just to verify that it 's giving us the same answer . so let 's say we have x squared plus 4x minus 21 is equal to 0 . so in this situation -- let me do that in a different color -- a is equal to 1 , right ? the coefficient on the x squared term is 1. b is equal to 4 , the coefficient on the x-term . and then c is equal to negative 21 , the constant term . and let 's just plug it in the formula , so what do we get ? we get x , this tells us that x is going to be equal to negative b . negative b is negative 4 -- i put the negative sign in front of that -- negative b plus or minus the square root of b squared . b squared is 16 , right ? 4 squared is 16 , minus 4 times a , which is 1 , times c , which is negative 21 . so we can put a 21 out there and that negative sign will cancel out just like that with that -- since this is the first time we 're doing it , let me not skip too many steps . so negative 21 , just so you can see how it fit in , and then all of that over 2a . a is 1 , so all of that over 2 . so what does this simplify , or hopefully it simplifies ? so we get x is equal to negative 4 plus or minus the square root of -- let 's see we have a negative times a negative , that 's going to give us a positive . and we had 16 plus , let 's see this is 6 , 4 times 1 is 4 times 21 is 84 . 16 plus 84 is 100 . that 's nice . that 's a nice perfect square . all of that over 2 , and so this is going to be equal to negative 4 plus or minus 10 over 2 . we could just divide both of these terms by 2 right now . so this is equal to negative 4 divided by 2 is negative 2 plus or minus 10 divided by 2 is 5 . so that tells us that x could be equal to negative 2 plus 5 , which is 3 , or x could be equal to negative 2 minus 5 , which is negative 7 . so the quadratic formula seems to have given us an answer for this . you can verify just by substituting back in that these do work , or you could even just try to factor this right here . you say what two numbers when you take their product , you get negative 21 and when you take their sum you get positive 4 ? so you 'd get x plus 7 times x minus 3 is equal to negative 21 . notice 7 times negative 3 is negative 21 , 7 minus 3 is positive 4 . you would get x plus -- sorry it 's not negative -- 21 is equal to 0 . there should be a 0 there . so you get x plus 7 is equal to 0 , or x minus 3 is equal to 0 . x could be equal to negative 7 or x could be equal to 3 . so it definitely gives us the same answer as factoring , so you might say , hey why bother with this crazy mess ? and the reason we want to bother with this crazy mess is it 'll also work for problems that are hard to factor . and let 's do a couple of those , let 's do some hard-to-factor problems right now . so let 's scroll down to get some fresh real estate . let 's rewrite the formula again , just in case we have n't had it memorized yet . x is going to be equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . i 'll supply this to another problem . let 's say we have the equation 3x squared plus 6x is equal to negative 10 . well , the first thing we want to do is get it in the form where all of our terms or on the left-hand side , so let 's add 10 to both sides of this equation . we get 3x squared plus the 6x plus 10 is equal to 0 . and now we can use a quadratic formula . so let 's apply it here . so a is equal to 3 . that is a , this is b and this right here is c. so the quadratic formula tells us the solutions to this equation . the roots of this quadratic function , i guess we could call it . x is going to be equal to negative b. b is 6 , so negative 6 plus or minus the square root of b squared . b is 6 , so we get 6 squared minus 4 times a , which is 3 times c , which is 10 . let 's stretch out the radical little bit , all of that over 2 times a , 2 times 3 . so we get x is equal to negative 6 plus or minus the square root of 36 minus -- this is interesting -- minus 4 times 3 times 10 . so this is minus -- 4 times 3 times 10 . so this is minus 120 . all of that over 6 . so this is interesting , you might already realize why it 's interesting . what is this going to simplify to ? 36 minus 120 is what ? that 's 84 . we make this into a 10 , this will become an 11 , this is a 4 . it is 84 , so this is going to be equal to negative 6 plus or minus the square root of -- but not positive 84 , that 's if it 's 120 minus 36 . we have 36 minus 120 . it 's going to be negative 84 all of that 6 . so you might say , gee , this is crazy . what a this silly quadratic formula you 're introducing me to , sal ? it 's worthless . it just gives me a square root of a negative number . it 's not giving me an answer . and the reason why it 's not giving you an answer , at least an answer that you might want , is because this will have no real solutions . in the future , we 're going to introduce something called an imaginary number , which is a square root of a negative number , and then we can actually express this in terms of those numbers . so this actually does have solutions , but they involve imaginary numbers . so this actually has no real solutions , we 're taking the square root of a negative number . so the b squared with the b squared minus 4ac , if this term right here is negative , then you 're not going to have any real solutions . and let 's verify that for ourselves . let 's get our graphic calculator out and let 's graph this equation right here . so , let 's get the graphs that y is equal to -- that 's what i had there before -- 3x squared plus 6x plus 10 . so that 's the equation and we 're going to see where it intersects the x-axis . where does it equal 0 ? so let me graph it . notice , this thing just comes down and then goes back up . its vertex is sitting here above the x-axis and it 's upward-opening . it never intersects the x-axis . so at no point will this expression , will this function , equal 0 . at no point will y equal 0 on this graph . so once again , the quadratic formula seems to be working . let 's do one more example , you can never see enough examples here . and i want to do ones that are , you know , maybe not so obvious to factor . so let 's say we get negative 3x squared plus 12x plus 1 is equal to 0 . now let 's try to do it just having the quadratic formula in our brain . so the x 's that satisfy this equation are going to be negative b . this is b so negative b is negative 12 plus or minus the square root of b squared , of 144 , that 's b squared minus 4 times a , which is negative 3 times c , which is 1 , all of that over 2 times a , over 2 times negative 3 . so all of that over negative 6 , this is going to be equal to negative 12 plus or minus the square root of -- what is this ? it 's a negative times a negative so they cancel out . so i have 144 plus 12 , so that is 156 , right ? 144 plus 12 , all of that over negative 6 . now , i suspect we can simplify this 156 . we could maybe bring some things out of the radical sign . so let 's attempt to do that . so let 's do a prime factorization of 156 . sometimes , this is the hardest part , simplifying the radical . so 156 is the same thing as 2 times 78 . 78 is the same thing as 2 times what ? that 's 2 times 39 . so the square root of 156 is equal to the square root of 2 times 2 times 39 or we could say that 's the square root of 2 times 2 times the square root of 39 . and this , obviously , is just going to be the square root of 4 or this is the square root of 2 times 2 is just 2 . 2 square roots of 39 , if i did that properly , let 's see , 4 times 39 . yeah , it looks like it 's right . so this up here will simplify to negative 12 plus or minus 2 times the square root of 39 , all of that over negative 6 . now we can divide the numerator and the denominator maybe by 2 . so this will be equal to negative 6 plus or minus the square root of 39 over negative 3 . or we could separate these two terms out . we could say this is equal to negative 6 over negative 3 plus or minus the square root of 39 over negative 3 . now , this is just a 2 right here , right ? these cancel out , 6 divided by 3 is 2 , so we get 2 . and now notice , if this is plus and we use this minus sign , the plus will become negative and the negative will become positive . but it still does n't matter , right ? we could say minus or plus , that 's the same thing as plus or minus the square root of 39 nine over 3 . i think that 's about as simple as we can get this answered . i want to make a very clear point of what i did that last step . i did not forget about this negative sign . i just said it does n't matter . it 's going to turn the positive into the negative ; it 's going to turn the negative into the positive . let me rewrite this . so this right here can be rewritten as 2 plus the square root of 39 over negative 3 or 2 minus the square root of 39 over negative 3 , right ? that 's what the plus or minus means , it could be this or that or both of them , really . now in this situation , this negative 3 will turn into 2 minus the square root of 39 over 3 , right ? i 'm just taking this negative out . here the negative and the negative will become a positive , and you get 2 plus the square root of 39 over 3 , right ? a negative times a negative is a positive . so once again , you have 2 plus or minus the square of 39 over 3 . 2 plus or minus the square root of 39 over 3 are solutions to this equation right there . let verify . i 'm just curious what the graph looks like . so let 's just look at it . let me clear this . where is the clear button ? so we have negative 3 three squared plus 12x plus 1 and let 's graph it . let 's see where it intersects the x-axis . it goes up there and then back down again . so 2 plus or minus the square , you see -- the square root of 39 is going to be a little bit more than 6 , right ? because 36 is 6 squared . so it 's going be a little bit more than 6 , so this is going to be a little bit more than 2 . a little bit more than 6 divided by 2 is a little bit more than 2 . so you 're going to get one value that 's a little bit more than 4 and then another value that should be a little bit less than 1 . and that looks like the case , you have 1 , 2 , 3 , 4 . you have a value that 's pretty close to 4 , and then you have another value that is a little bit -- it looks close to 0 but maybe a little bit less than that . so anyway , hopefully you found this application of the quadratic formula helpful .
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you say what two numbers when you take their product , you get negative 21 and when you take their sum you get positive 4 ? so you 'd get x plus 7 times x minus 3 is equal to negative 21 . notice 7 times negative 3 is negative 21 , 7 minus 3 is positive 4 .
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does the quadric formula work if the x is say x^4 or x^x^2 ?
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in this video , i 'm going to expose you to what is maybe one of at least the top five most useful formulas in mathematics . and if you 've seen many of my videos , you know that i 'm not a big fan of memorizing things . but i will recommend you memorize it with the caveat that you also remember how to prove it , because i do n't want you to just remember things and not know where they came from . but with that said , let me show you what i 'm talking about : it 's the quadratic formula . and as you might guess , it is to solve for the roots , or the zeroes of quadratic equations . so let 's speak in very general terms and i 'll show you some examples . so let 's say i have an equation of the form ax squared plus bx plus c is equal to 0 . you should recognize this . this is a quadratic equation where a , b and c are -- well , a is the coefficient on the x squared term or the second degree term , b is the coefficient on the x term and then c , is , you could imagine , the coefficient on the x to the zero term , or it 's the constant term . now , given that you have a general quadratic equation like this , the quadratic formula tells us that the solutions to this equation are x is equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . and i know it seems crazy and convoluted and hard for you to memorize right now , but as you get a lot more practice you 'll see that it actually is a pretty reasonable formula to stick in your brain someplace . and you might say , gee , this is a wacky formula , where did it come from ? and in the next video i 'm going to show you where it came from . but i want you to get used to using it first . but it really just came from completing the square on this equation right there . if you complete the square here , you 're actually going to get this solution and that is the quadratic formula , right there . so let 's apply it to some problems . let 's start off with something that we could have factored just to verify that it 's giving us the same answer . so let 's say we have x squared plus 4x minus 21 is equal to 0 . so in this situation -- let me do that in a different color -- a is equal to 1 , right ? the coefficient on the x squared term is 1. b is equal to 4 , the coefficient on the x-term . and then c is equal to negative 21 , the constant term . and let 's just plug it in the formula , so what do we get ? we get x , this tells us that x is going to be equal to negative b . negative b is negative 4 -- i put the negative sign in front of that -- negative b plus or minus the square root of b squared . b squared is 16 , right ? 4 squared is 16 , minus 4 times a , which is 1 , times c , which is negative 21 . so we can put a 21 out there and that negative sign will cancel out just like that with that -- since this is the first time we 're doing it , let me not skip too many steps . so negative 21 , just so you can see how it fit in , and then all of that over 2a . a is 1 , so all of that over 2 . so what does this simplify , or hopefully it simplifies ? so we get x is equal to negative 4 plus or minus the square root of -- let 's see we have a negative times a negative , that 's going to give us a positive . and we had 16 plus , let 's see this is 6 , 4 times 1 is 4 times 21 is 84 . 16 plus 84 is 100 . that 's nice . that 's a nice perfect square . all of that over 2 , and so this is going to be equal to negative 4 plus or minus 10 over 2 . we could just divide both of these terms by 2 right now . so this is equal to negative 4 divided by 2 is negative 2 plus or minus 10 divided by 2 is 5 . so that tells us that x could be equal to negative 2 plus 5 , which is 3 , or x could be equal to negative 2 minus 5 , which is negative 7 . so the quadratic formula seems to have given us an answer for this . you can verify just by substituting back in that these do work , or you could even just try to factor this right here . you say what two numbers when you take their product , you get negative 21 and when you take their sum you get positive 4 ? so you 'd get x plus 7 times x minus 3 is equal to negative 21 . notice 7 times negative 3 is negative 21 , 7 minus 3 is positive 4 . you would get x plus -- sorry it 's not negative -- 21 is equal to 0 . there should be a 0 there . so you get x plus 7 is equal to 0 , or x minus 3 is equal to 0 . x could be equal to negative 7 or x could be equal to 3 . so it definitely gives us the same answer as factoring , so you might say , hey why bother with this crazy mess ? and the reason we want to bother with this crazy mess is it 'll also work for problems that are hard to factor . and let 's do a couple of those , let 's do some hard-to-factor problems right now . so let 's scroll down to get some fresh real estate . let 's rewrite the formula again , just in case we have n't had it memorized yet . x is going to be equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . i 'll supply this to another problem . let 's say we have the equation 3x squared plus 6x is equal to negative 10 . well , the first thing we want to do is get it in the form where all of our terms or on the left-hand side , so let 's add 10 to both sides of this equation . we get 3x squared plus the 6x plus 10 is equal to 0 . and now we can use a quadratic formula . so let 's apply it here . so a is equal to 3 . that is a , this is b and this right here is c. so the quadratic formula tells us the solutions to this equation . the roots of this quadratic function , i guess we could call it . x is going to be equal to negative b. b is 6 , so negative 6 plus or minus the square root of b squared . b is 6 , so we get 6 squared minus 4 times a , which is 3 times c , which is 10 . let 's stretch out the radical little bit , all of that over 2 times a , 2 times 3 . so we get x is equal to negative 6 plus or minus the square root of 36 minus -- this is interesting -- minus 4 times 3 times 10 . so this is minus -- 4 times 3 times 10 . so this is minus 120 . all of that over 6 . so this is interesting , you might already realize why it 's interesting . what is this going to simplify to ? 36 minus 120 is what ? that 's 84 . we make this into a 10 , this will become an 11 , this is a 4 . it is 84 , so this is going to be equal to negative 6 plus or minus the square root of -- but not positive 84 , that 's if it 's 120 minus 36 . we have 36 minus 120 . it 's going to be negative 84 all of that 6 . so you might say , gee , this is crazy . what a this silly quadratic formula you 're introducing me to , sal ? it 's worthless . it just gives me a square root of a negative number . it 's not giving me an answer . and the reason why it 's not giving you an answer , at least an answer that you might want , is because this will have no real solutions . in the future , we 're going to introduce something called an imaginary number , which is a square root of a negative number , and then we can actually express this in terms of those numbers . so this actually does have solutions , but they involve imaginary numbers . so this actually has no real solutions , we 're taking the square root of a negative number . so the b squared with the b squared minus 4ac , if this term right here is negative , then you 're not going to have any real solutions . and let 's verify that for ourselves . let 's get our graphic calculator out and let 's graph this equation right here . so , let 's get the graphs that y is equal to -- that 's what i had there before -- 3x squared plus 6x plus 10 . so that 's the equation and we 're going to see where it intersects the x-axis . where does it equal 0 ? so let me graph it . notice , this thing just comes down and then goes back up . its vertex is sitting here above the x-axis and it 's upward-opening . it never intersects the x-axis . so at no point will this expression , will this function , equal 0 . at no point will y equal 0 on this graph . so once again , the quadratic formula seems to be working . let 's do one more example , you can never see enough examples here . and i want to do ones that are , you know , maybe not so obvious to factor . so let 's say we get negative 3x squared plus 12x plus 1 is equal to 0 . now let 's try to do it just having the quadratic formula in our brain . so the x 's that satisfy this equation are going to be negative b . this is b so negative b is negative 12 plus or minus the square root of b squared , of 144 , that 's b squared minus 4 times a , which is negative 3 times c , which is 1 , all of that over 2 times a , over 2 times negative 3 . so all of that over negative 6 , this is going to be equal to negative 12 plus or minus the square root of -- what is this ? it 's a negative times a negative so they cancel out . so i have 144 plus 12 , so that is 156 , right ? 144 plus 12 , all of that over negative 6 . now , i suspect we can simplify this 156 . we could maybe bring some things out of the radical sign . so let 's attempt to do that . so let 's do a prime factorization of 156 . sometimes , this is the hardest part , simplifying the radical . so 156 is the same thing as 2 times 78 . 78 is the same thing as 2 times what ? that 's 2 times 39 . so the square root of 156 is equal to the square root of 2 times 2 times 39 or we could say that 's the square root of 2 times 2 times the square root of 39 . and this , obviously , is just going to be the square root of 4 or this is the square root of 2 times 2 is just 2 . 2 square roots of 39 , if i did that properly , let 's see , 4 times 39 . yeah , it looks like it 's right . so this up here will simplify to negative 12 plus or minus 2 times the square root of 39 , all of that over negative 6 . now we can divide the numerator and the denominator maybe by 2 . so this will be equal to negative 6 plus or minus the square root of 39 over negative 3 . or we could separate these two terms out . we could say this is equal to negative 6 over negative 3 plus or minus the square root of 39 over negative 3 . now , this is just a 2 right here , right ? these cancel out , 6 divided by 3 is 2 , so we get 2 . and now notice , if this is plus and we use this minus sign , the plus will become negative and the negative will become positive . but it still does n't matter , right ? we could say minus or plus , that 's the same thing as plus or minus the square root of 39 nine over 3 . i think that 's about as simple as we can get this answered . i want to make a very clear point of what i did that last step . i did not forget about this negative sign . i just said it does n't matter . it 's going to turn the positive into the negative ; it 's going to turn the negative into the positive . let me rewrite this . so this right here can be rewritten as 2 plus the square root of 39 over negative 3 or 2 minus the square root of 39 over negative 3 , right ? that 's what the plus or minus means , it could be this or that or both of them , really . now in this situation , this negative 3 will turn into 2 minus the square root of 39 over 3 , right ? i 'm just taking this negative out . here the negative and the negative will become a positive , and you get 2 plus the square root of 39 over 3 , right ? a negative times a negative is a positive . so once again , you have 2 plus or minus the square of 39 over 3 . 2 plus or minus the square root of 39 over 3 are solutions to this equation right there . let verify . i 'm just curious what the graph looks like . so let 's just look at it . let me clear this . where is the clear button ? so we have negative 3 three squared plus 12x plus 1 and let 's graph it . let 's see where it intersects the x-axis . it goes up there and then back down again . so 2 plus or minus the square , you see -- the square root of 39 is going to be a little bit more than 6 , right ? because 36 is 6 squared . so it 's going be a little bit more than 6 , so this is going to be a little bit more than 2 . a little bit more than 6 divided by 2 is a little bit more than 2 . so you 're going to get one value that 's a little bit more than 4 and then another value that should be a little bit less than 1 . and that looks like the case , you have 1 , 2 , 3 , 4 . you have a value that 's pretty close to 4 , and then you have another value that is a little bit -- it looks close to 0 but maybe a little bit less than that . so anyway , hopefully you found this application of the quadratic formula helpful .
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notice 7 times negative 3 is negative 21 , 7 minus 3 is positive 4 . you would get x plus -- sorry it 's not negative -- 21 is equal to 0 . there should be a 0 there .
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lets say the problem is x^2-225=0 would the quadratic formula still be able to work for a missing b value ?
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in this video , i 'm going to expose you to what is maybe one of at least the top five most useful formulas in mathematics . and if you 've seen many of my videos , you know that i 'm not a big fan of memorizing things . but i will recommend you memorize it with the caveat that you also remember how to prove it , because i do n't want you to just remember things and not know where they came from . but with that said , let me show you what i 'm talking about : it 's the quadratic formula . and as you might guess , it is to solve for the roots , or the zeroes of quadratic equations . so let 's speak in very general terms and i 'll show you some examples . so let 's say i have an equation of the form ax squared plus bx plus c is equal to 0 . you should recognize this . this is a quadratic equation where a , b and c are -- well , a is the coefficient on the x squared term or the second degree term , b is the coefficient on the x term and then c , is , you could imagine , the coefficient on the x to the zero term , or it 's the constant term . now , given that you have a general quadratic equation like this , the quadratic formula tells us that the solutions to this equation are x is equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . and i know it seems crazy and convoluted and hard for you to memorize right now , but as you get a lot more practice you 'll see that it actually is a pretty reasonable formula to stick in your brain someplace . and you might say , gee , this is a wacky formula , where did it come from ? and in the next video i 'm going to show you where it came from . but i want you to get used to using it first . but it really just came from completing the square on this equation right there . if you complete the square here , you 're actually going to get this solution and that is the quadratic formula , right there . so let 's apply it to some problems . let 's start off with something that we could have factored just to verify that it 's giving us the same answer . so let 's say we have x squared plus 4x minus 21 is equal to 0 . so in this situation -- let me do that in a different color -- a is equal to 1 , right ? the coefficient on the x squared term is 1. b is equal to 4 , the coefficient on the x-term . and then c is equal to negative 21 , the constant term . and let 's just plug it in the formula , so what do we get ? we get x , this tells us that x is going to be equal to negative b . negative b is negative 4 -- i put the negative sign in front of that -- negative b plus or minus the square root of b squared . b squared is 16 , right ? 4 squared is 16 , minus 4 times a , which is 1 , times c , which is negative 21 . so we can put a 21 out there and that negative sign will cancel out just like that with that -- since this is the first time we 're doing it , let me not skip too many steps . so negative 21 , just so you can see how it fit in , and then all of that over 2a . a is 1 , so all of that over 2 . so what does this simplify , or hopefully it simplifies ? so we get x is equal to negative 4 plus or minus the square root of -- let 's see we have a negative times a negative , that 's going to give us a positive . and we had 16 plus , let 's see this is 6 , 4 times 1 is 4 times 21 is 84 . 16 plus 84 is 100 . that 's nice . that 's a nice perfect square . all of that over 2 , and so this is going to be equal to negative 4 plus or minus 10 over 2 . we could just divide both of these terms by 2 right now . so this is equal to negative 4 divided by 2 is negative 2 plus or minus 10 divided by 2 is 5 . so that tells us that x could be equal to negative 2 plus 5 , which is 3 , or x could be equal to negative 2 minus 5 , which is negative 7 . so the quadratic formula seems to have given us an answer for this . you can verify just by substituting back in that these do work , or you could even just try to factor this right here . you say what two numbers when you take their product , you get negative 21 and when you take their sum you get positive 4 ? so you 'd get x plus 7 times x minus 3 is equal to negative 21 . notice 7 times negative 3 is negative 21 , 7 minus 3 is positive 4 . you would get x plus -- sorry it 's not negative -- 21 is equal to 0 . there should be a 0 there . so you get x plus 7 is equal to 0 , or x minus 3 is equal to 0 . x could be equal to negative 7 or x could be equal to 3 . so it definitely gives us the same answer as factoring , so you might say , hey why bother with this crazy mess ? and the reason we want to bother with this crazy mess is it 'll also work for problems that are hard to factor . and let 's do a couple of those , let 's do some hard-to-factor problems right now . so let 's scroll down to get some fresh real estate . let 's rewrite the formula again , just in case we have n't had it memorized yet . x is going to be equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . i 'll supply this to another problem . let 's say we have the equation 3x squared plus 6x is equal to negative 10 . well , the first thing we want to do is get it in the form where all of our terms or on the left-hand side , so let 's add 10 to both sides of this equation . we get 3x squared plus the 6x plus 10 is equal to 0 . and now we can use a quadratic formula . so let 's apply it here . so a is equal to 3 . that is a , this is b and this right here is c. so the quadratic formula tells us the solutions to this equation . the roots of this quadratic function , i guess we could call it . x is going to be equal to negative b. b is 6 , so negative 6 plus or minus the square root of b squared . b is 6 , so we get 6 squared minus 4 times a , which is 3 times c , which is 10 . let 's stretch out the radical little bit , all of that over 2 times a , 2 times 3 . so we get x is equal to negative 6 plus or minus the square root of 36 minus -- this is interesting -- minus 4 times 3 times 10 . so this is minus -- 4 times 3 times 10 . so this is minus 120 . all of that over 6 . so this is interesting , you might already realize why it 's interesting . what is this going to simplify to ? 36 minus 120 is what ? that 's 84 . we make this into a 10 , this will become an 11 , this is a 4 . it is 84 , so this is going to be equal to negative 6 plus or minus the square root of -- but not positive 84 , that 's if it 's 120 minus 36 . we have 36 minus 120 . it 's going to be negative 84 all of that 6 . so you might say , gee , this is crazy . what a this silly quadratic formula you 're introducing me to , sal ? it 's worthless . it just gives me a square root of a negative number . it 's not giving me an answer . and the reason why it 's not giving you an answer , at least an answer that you might want , is because this will have no real solutions . in the future , we 're going to introduce something called an imaginary number , which is a square root of a negative number , and then we can actually express this in terms of those numbers . so this actually does have solutions , but they involve imaginary numbers . so this actually has no real solutions , we 're taking the square root of a negative number . so the b squared with the b squared minus 4ac , if this term right here is negative , then you 're not going to have any real solutions . and let 's verify that for ourselves . let 's get our graphic calculator out and let 's graph this equation right here . so , let 's get the graphs that y is equal to -- that 's what i had there before -- 3x squared plus 6x plus 10 . so that 's the equation and we 're going to see where it intersects the x-axis . where does it equal 0 ? so let me graph it . notice , this thing just comes down and then goes back up . its vertex is sitting here above the x-axis and it 's upward-opening . it never intersects the x-axis . so at no point will this expression , will this function , equal 0 . at no point will y equal 0 on this graph . so once again , the quadratic formula seems to be working . let 's do one more example , you can never see enough examples here . and i want to do ones that are , you know , maybe not so obvious to factor . so let 's say we get negative 3x squared plus 12x plus 1 is equal to 0 . now let 's try to do it just having the quadratic formula in our brain . so the x 's that satisfy this equation are going to be negative b . this is b so negative b is negative 12 plus or minus the square root of b squared , of 144 , that 's b squared minus 4 times a , which is negative 3 times c , which is 1 , all of that over 2 times a , over 2 times negative 3 . so all of that over negative 6 , this is going to be equal to negative 12 plus or minus the square root of -- what is this ? it 's a negative times a negative so they cancel out . so i have 144 plus 12 , so that is 156 , right ? 144 plus 12 , all of that over negative 6 . now , i suspect we can simplify this 156 . we could maybe bring some things out of the radical sign . so let 's attempt to do that . so let 's do a prime factorization of 156 . sometimes , this is the hardest part , simplifying the radical . so 156 is the same thing as 2 times 78 . 78 is the same thing as 2 times what ? that 's 2 times 39 . so the square root of 156 is equal to the square root of 2 times 2 times 39 or we could say that 's the square root of 2 times 2 times the square root of 39 . and this , obviously , is just going to be the square root of 4 or this is the square root of 2 times 2 is just 2 . 2 square roots of 39 , if i did that properly , let 's see , 4 times 39 . yeah , it looks like it 's right . so this up here will simplify to negative 12 plus or minus 2 times the square root of 39 , all of that over negative 6 . now we can divide the numerator and the denominator maybe by 2 . so this will be equal to negative 6 plus or minus the square root of 39 over negative 3 . or we could separate these two terms out . we could say this is equal to negative 6 over negative 3 plus or minus the square root of 39 over negative 3 . now , this is just a 2 right here , right ? these cancel out , 6 divided by 3 is 2 , so we get 2 . and now notice , if this is plus and we use this minus sign , the plus will become negative and the negative will become positive . but it still does n't matter , right ? we could say minus or plus , that 's the same thing as plus or minus the square root of 39 nine over 3 . i think that 's about as simple as we can get this answered . i want to make a very clear point of what i did that last step . i did not forget about this negative sign . i just said it does n't matter . it 's going to turn the positive into the negative ; it 's going to turn the negative into the positive . let me rewrite this . so this right here can be rewritten as 2 plus the square root of 39 over negative 3 or 2 minus the square root of 39 over negative 3 , right ? that 's what the plus or minus means , it could be this or that or both of them , really . now in this situation , this negative 3 will turn into 2 minus the square root of 39 over 3 , right ? i 'm just taking this negative out . here the negative and the negative will become a positive , and you get 2 plus the square root of 39 over 3 , right ? a negative times a negative is a positive . so once again , you have 2 plus or minus the square of 39 over 3 . 2 plus or minus the square root of 39 over 3 are solutions to this equation right there . let verify . i 'm just curious what the graph looks like . so let 's just look at it . let me clear this . where is the clear button ? so we have negative 3 three squared plus 12x plus 1 and let 's graph it . let 's see where it intersects the x-axis . it goes up there and then back down again . so 2 plus or minus the square , you see -- the square root of 39 is going to be a little bit more than 6 , right ? because 36 is 6 squared . so it 's going be a little bit more than 6 , so this is going to be a little bit more than 2 . a little bit more than 6 divided by 2 is a little bit more than 2 . so you 're going to get one value that 's a little bit more than 4 and then another value that should be a little bit less than 1 . and that looks like the case , you have 1 , 2 , 3 , 4 . you have a value that 's pretty close to 4 , and then you have another value that is a little bit -- it looks close to 0 but maybe a little bit less than that . so anyway , hopefully you found this application of the quadratic formula helpful .
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so let 's speak in very general terms and i 'll show you some examples . so let 's say i have an equation of the form ax squared plus bx plus c is equal to 0 . you should recognize this .
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what is the surd form ?
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in this video , i 'm going to expose you to what is maybe one of at least the top five most useful formulas in mathematics . and if you 've seen many of my videos , you know that i 'm not a big fan of memorizing things . but i will recommend you memorize it with the caveat that you also remember how to prove it , because i do n't want you to just remember things and not know where they came from . but with that said , let me show you what i 'm talking about : it 's the quadratic formula . and as you might guess , it is to solve for the roots , or the zeroes of quadratic equations . so let 's speak in very general terms and i 'll show you some examples . so let 's say i have an equation of the form ax squared plus bx plus c is equal to 0 . you should recognize this . this is a quadratic equation where a , b and c are -- well , a is the coefficient on the x squared term or the second degree term , b is the coefficient on the x term and then c , is , you could imagine , the coefficient on the x to the zero term , or it 's the constant term . now , given that you have a general quadratic equation like this , the quadratic formula tells us that the solutions to this equation are x is equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . and i know it seems crazy and convoluted and hard for you to memorize right now , but as you get a lot more practice you 'll see that it actually is a pretty reasonable formula to stick in your brain someplace . and you might say , gee , this is a wacky formula , where did it come from ? and in the next video i 'm going to show you where it came from . but i want you to get used to using it first . but it really just came from completing the square on this equation right there . if you complete the square here , you 're actually going to get this solution and that is the quadratic formula , right there . so let 's apply it to some problems . let 's start off with something that we could have factored just to verify that it 's giving us the same answer . so let 's say we have x squared plus 4x minus 21 is equal to 0 . so in this situation -- let me do that in a different color -- a is equal to 1 , right ? the coefficient on the x squared term is 1. b is equal to 4 , the coefficient on the x-term . and then c is equal to negative 21 , the constant term . and let 's just plug it in the formula , so what do we get ? we get x , this tells us that x is going to be equal to negative b . negative b is negative 4 -- i put the negative sign in front of that -- negative b plus or minus the square root of b squared . b squared is 16 , right ? 4 squared is 16 , minus 4 times a , which is 1 , times c , which is negative 21 . so we can put a 21 out there and that negative sign will cancel out just like that with that -- since this is the first time we 're doing it , let me not skip too many steps . so negative 21 , just so you can see how it fit in , and then all of that over 2a . a is 1 , so all of that over 2 . so what does this simplify , or hopefully it simplifies ? so we get x is equal to negative 4 plus or minus the square root of -- let 's see we have a negative times a negative , that 's going to give us a positive . and we had 16 plus , let 's see this is 6 , 4 times 1 is 4 times 21 is 84 . 16 plus 84 is 100 . that 's nice . that 's a nice perfect square . all of that over 2 , and so this is going to be equal to negative 4 plus or minus 10 over 2 . we could just divide both of these terms by 2 right now . so this is equal to negative 4 divided by 2 is negative 2 plus or minus 10 divided by 2 is 5 . so that tells us that x could be equal to negative 2 plus 5 , which is 3 , or x could be equal to negative 2 minus 5 , which is negative 7 . so the quadratic formula seems to have given us an answer for this . you can verify just by substituting back in that these do work , or you could even just try to factor this right here . you say what two numbers when you take their product , you get negative 21 and when you take their sum you get positive 4 ? so you 'd get x plus 7 times x minus 3 is equal to negative 21 . notice 7 times negative 3 is negative 21 , 7 minus 3 is positive 4 . you would get x plus -- sorry it 's not negative -- 21 is equal to 0 . there should be a 0 there . so you get x plus 7 is equal to 0 , or x minus 3 is equal to 0 . x could be equal to negative 7 or x could be equal to 3 . so it definitely gives us the same answer as factoring , so you might say , hey why bother with this crazy mess ? and the reason we want to bother with this crazy mess is it 'll also work for problems that are hard to factor . and let 's do a couple of those , let 's do some hard-to-factor problems right now . so let 's scroll down to get some fresh real estate . let 's rewrite the formula again , just in case we have n't had it memorized yet . x is going to be equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . i 'll supply this to another problem . let 's say we have the equation 3x squared plus 6x is equal to negative 10 . well , the first thing we want to do is get it in the form where all of our terms or on the left-hand side , so let 's add 10 to both sides of this equation . we get 3x squared plus the 6x plus 10 is equal to 0 . and now we can use a quadratic formula . so let 's apply it here . so a is equal to 3 . that is a , this is b and this right here is c. so the quadratic formula tells us the solutions to this equation . the roots of this quadratic function , i guess we could call it . x is going to be equal to negative b. b is 6 , so negative 6 plus or minus the square root of b squared . b is 6 , so we get 6 squared minus 4 times a , which is 3 times c , which is 10 . let 's stretch out the radical little bit , all of that over 2 times a , 2 times 3 . so we get x is equal to negative 6 plus or minus the square root of 36 minus -- this is interesting -- minus 4 times 3 times 10 . so this is minus -- 4 times 3 times 10 . so this is minus 120 . all of that over 6 . so this is interesting , you might already realize why it 's interesting . what is this going to simplify to ? 36 minus 120 is what ? that 's 84 . we make this into a 10 , this will become an 11 , this is a 4 . it is 84 , so this is going to be equal to negative 6 plus or minus the square root of -- but not positive 84 , that 's if it 's 120 minus 36 . we have 36 minus 120 . it 's going to be negative 84 all of that 6 . so you might say , gee , this is crazy . what a this silly quadratic formula you 're introducing me to , sal ? it 's worthless . it just gives me a square root of a negative number . it 's not giving me an answer . and the reason why it 's not giving you an answer , at least an answer that you might want , is because this will have no real solutions . in the future , we 're going to introduce something called an imaginary number , which is a square root of a negative number , and then we can actually express this in terms of those numbers . so this actually does have solutions , but they involve imaginary numbers . so this actually has no real solutions , we 're taking the square root of a negative number . so the b squared with the b squared minus 4ac , if this term right here is negative , then you 're not going to have any real solutions . and let 's verify that for ourselves . let 's get our graphic calculator out and let 's graph this equation right here . so , let 's get the graphs that y is equal to -- that 's what i had there before -- 3x squared plus 6x plus 10 . so that 's the equation and we 're going to see where it intersects the x-axis . where does it equal 0 ? so let me graph it . notice , this thing just comes down and then goes back up . its vertex is sitting here above the x-axis and it 's upward-opening . it never intersects the x-axis . so at no point will this expression , will this function , equal 0 . at no point will y equal 0 on this graph . so once again , the quadratic formula seems to be working . let 's do one more example , you can never see enough examples here . and i want to do ones that are , you know , maybe not so obvious to factor . so let 's say we get negative 3x squared plus 12x plus 1 is equal to 0 . now let 's try to do it just having the quadratic formula in our brain . so the x 's that satisfy this equation are going to be negative b . this is b so negative b is negative 12 plus or minus the square root of b squared , of 144 , that 's b squared minus 4 times a , which is negative 3 times c , which is 1 , all of that over 2 times a , over 2 times negative 3 . so all of that over negative 6 , this is going to be equal to negative 12 plus or minus the square root of -- what is this ? it 's a negative times a negative so they cancel out . so i have 144 plus 12 , so that is 156 , right ? 144 plus 12 , all of that over negative 6 . now , i suspect we can simplify this 156 . we could maybe bring some things out of the radical sign . so let 's attempt to do that . so let 's do a prime factorization of 156 . sometimes , this is the hardest part , simplifying the radical . so 156 is the same thing as 2 times 78 . 78 is the same thing as 2 times what ? that 's 2 times 39 . so the square root of 156 is equal to the square root of 2 times 2 times 39 or we could say that 's the square root of 2 times 2 times the square root of 39 . and this , obviously , is just going to be the square root of 4 or this is the square root of 2 times 2 is just 2 . 2 square roots of 39 , if i did that properly , let 's see , 4 times 39 . yeah , it looks like it 's right . so this up here will simplify to negative 12 plus or minus 2 times the square root of 39 , all of that over negative 6 . now we can divide the numerator and the denominator maybe by 2 . so this will be equal to negative 6 plus or minus the square root of 39 over negative 3 . or we could separate these two terms out . we could say this is equal to negative 6 over negative 3 plus or minus the square root of 39 over negative 3 . now , this is just a 2 right here , right ? these cancel out , 6 divided by 3 is 2 , so we get 2 . and now notice , if this is plus and we use this minus sign , the plus will become negative and the negative will become positive . but it still does n't matter , right ? we could say minus or plus , that 's the same thing as plus or minus the square root of 39 nine over 3 . i think that 's about as simple as we can get this answered . i want to make a very clear point of what i did that last step . i did not forget about this negative sign . i just said it does n't matter . it 's going to turn the positive into the negative ; it 's going to turn the negative into the positive . let me rewrite this . so this right here can be rewritten as 2 plus the square root of 39 over negative 3 or 2 minus the square root of 39 over negative 3 , right ? that 's what the plus or minus means , it could be this or that or both of them , really . now in this situation , this negative 3 will turn into 2 minus the square root of 39 over 3 , right ? i 'm just taking this negative out . here the negative and the negative will become a positive , and you get 2 plus the square root of 39 over 3 , right ? a negative times a negative is a positive . so once again , you have 2 plus or minus the square of 39 over 3 . 2 plus or minus the square root of 39 over 3 are solutions to this equation right there . let verify . i 'm just curious what the graph looks like . so let 's just look at it . let me clear this . where is the clear button ? so we have negative 3 three squared plus 12x plus 1 and let 's graph it . let 's see where it intersects the x-axis . it goes up there and then back down again . so 2 plus or minus the square , you see -- the square root of 39 is going to be a little bit more than 6 , right ? because 36 is 6 squared . so it 's going be a little bit more than 6 , so this is going to be a little bit more than 2 . a little bit more than 6 divided by 2 is a little bit more than 2 . so you 're going to get one value that 's a little bit more than 4 and then another value that should be a little bit less than 1 . and that looks like the case , you have 1 , 2 , 3 , 4 . you have a value that 's pretty close to 4 , and then you have another value that is a little bit -- it looks close to 0 but maybe a little bit less than that . so anyway , hopefully you found this application of the quadratic formula helpful .
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we get x , this tells us that x is going to be equal to negative b . negative b is negative 4 -- i put the negative sign in front of that -- negative b plus or minus the square root of b squared . b squared is 16 , right ?
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if the b is already negative do you put it in the formula ?
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in this video , i 'm going to expose you to what is maybe one of at least the top five most useful formulas in mathematics . and if you 've seen many of my videos , you know that i 'm not a big fan of memorizing things . but i will recommend you memorize it with the caveat that you also remember how to prove it , because i do n't want you to just remember things and not know where they came from . but with that said , let me show you what i 'm talking about : it 's the quadratic formula . and as you might guess , it is to solve for the roots , or the zeroes of quadratic equations . so let 's speak in very general terms and i 'll show you some examples . so let 's say i have an equation of the form ax squared plus bx plus c is equal to 0 . you should recognize this . this is a quadratic equation where a , b and c are -- well , a is the coefficient on the x squared term or the second degree term , b is the coefficient on the x term and then c , is , you could imagine , the coefficient on the x to the zero term , or it 's the constant term . now , given that you have a general quadratic equation like this , the quadratic formula tells us that the solutions to this equation are x is equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . and i know it seems crazy and convoluted and hard for you to memorize right now , but as you get a lot more practice you 'll see that it actually is a pretty reasonable formula to stick in your brain someplace . and you might say , gee , this is a wacky formula , where did it come from ? and in the next video i 'm going to show you where it came from . but i want you to get used to using it first . but it really just came from completing the square on this equation right there . if you complete the square here , you 're actually going to get this solution and that is the quadratic formula , right there . so let 's apply it to some problems . let 's start off with something that we could have factored just to verify that it 's giving us the same answer . so let 's say we have x squared plus 4x minus 21 is equal to 0 . so in this situation -- let me do that in a different color -- a is equal to 1 , right ? the coefficient on the x squared term is 1. b is equal to 4 , the coefficient on the x-term . and then c is equal to negative 21 , the constant term . and let 's just plug it in the formula , so what do we get ? we get x , this tells us that x is going to be equal to negative b . negative b is negative 4 -- i put the negative sign in front of that -- negative b plus or minus the square root of b squared . b squared is 16 , right ? 4 squared is 16 , minus 4 times a , which is 1 , times c , which is negative 21 . so we can put a 21 out there and that negative sign will cancel out just like that with that -- since this is the first time we 're doing it , let me not skip too many steps . so negative 21 , just so you can see how it fit in , and then all of that over 2a . a is 1 , so all of that over 2 . so what does this simplify , or hopefully it simplifies ? so we get x is equal to negative 4 plus or minus the square root of -- let 's see we have a negative times a negative , that 's going to give us a positive . and we had 16 plus , let 's see this is 6 , 4 times 1 is 4 times 21 is 84 . 16 plus 84 is 100 . that 's nice . that 's a nice perfect square . all of that over 2 , and so this is going to be equal to negative 4 plus or minus 10 over 2 . we could just divide both of these terms by 2 right now . so this is equal to negative 4 divided by 2 is negative 2 plus or minus 10 divided by 2 is 5 . so that tells us that x could be equal to negative 2 plus 5 , which is 3 , or x could be equal to negative 2 minus 5 , which is negative 7 . so the quadratic formula seems to have given us an answer for this . you can verify just by substituting back in that these do work , or you could even just try to factor this right here . you say what two numbers when you take their product , you get negative 21 and when you take their sum you get positive 4 ? so you 'd get x plus 7 times x minus 3 is equal to negative 21 . notice 7 times negative 3 is negative 21 , 7 minus 3 is positive 4 . you would get x plus -- sorry it 's not negative -- 21 is equal to 0 . there should be a 0 there . so you get x plus 7 is equal to 0 , or x minus 3 is equal to 0 . x could be equal to negative 7 or x could be equal to 3 . so it definitely gives us the same answer as factoring , so you might say , hey why bother with this crazy mess ? and the reason we want to bother with this crazy mess is it 'll also work for problems that are hard to factor . and let 's do a couple of those , let 's do some hard-to-factor problems right now . so let 's scroll down to get some fresh real estate . let 's rewrite the formula again , just in case we have n't had it memorized yet . x is going to be equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . i 'll supply this to another problem . let 's say we have the equation 3x squared plus 6x is equal to negative 10 . well , the first thing we want to do is get it in the form where all of our terms or on the left-hand side , so let 's add 10 to both sides of this equation . we get 3x squared plus the 6x plus 10 is equal to 0 . and now we can use a quadratic formula . so let 's apply it here . so a is equal to 3 . that is a , this is b and this right here is c. so the quadratic formula tells us the solutions to this equation . the roots of this quadratic function , i guess we could call it . x is going to be equal to negative b. b is 6 , so negative 6 plus or minus the square root of b squared . b is 6 , so we get 6 squared minus 4 times a , which is 3 times c , which is 10 . let 's stretch out the radical little bit , all of that over 2 times a , 2 times 3 . so we get x is equal to negative 6 plus or minus the square root of 36 minus -- this is interesting -- minus 4 times 3 times 10 . so this is minus -- 4 times 3 times 10 . so this is minus 120 . all of that over 6 . so this is interesting , you might already realize why it 's interesting . what is this going to simplify to ? 36 minus 120 is what ? that 's 84 . we make this into a 10 , this will become an 11 , this is a 4 . it is 84 , so this is going to be equal to negative 6 plus or minus the square root of -- but not positive 84 , that 's if it 's 120 minus 36 . we have 36 minus 120 . it 's going to be negative 84 all of that 6 . so you might say , gee , this is crazy . what a this silly quadratic formula you 're introducing me to , sal ? it 's worthless . it just gives me a square root of a negative number . it 's not giving me an answer . and the reason why it 's not giving you an answer , at least an answer that you might want , is because this will have no real solutions . in the future , we 're going to introduce something called an imaginary number , which is a square root of a negative number , and then we can actually express this in terms of those numbers . so this actually does have solutions , but they involve imaginary numbers . so this actually has no real solutions , we 're taking the square root of a negative number . so the b squared with the b squared minus 4ac , if this term right here is negative , then you 're not going to have any real solutions . and let 's verify that for ourselves . let 's get our graphic calculator out and let 's graph this equation right here . so , let 's get the graphs that y is equal to -- that 's what i had there before -- 3x squared plus 6x plus 10 . so that 's the equation and we 're going to see where it intersects the x-axis . where does it equal 0 ? so let me graph it . notice , this thing just comes down and then goes back up . its vertex is sitting here above the x-axis and it 's upward-opening . it never intersects the x-axis . so at no point will this expression , will this function , equal 0 . at no point will y equal 0 on this graph . so once again , the quadratic formula seems to be working . let 's do one more example , you can never see enough examples here . and i want to do ones that are , you know , maybe not so obvious to factor . so let 's say we get negative 3x squared plus 12x plus 1 is equal to 0 . now let 's try to do it just having the quadratic formula in our brain . so the x 's that satisfy this equation are going to be negative b . this is b so negative b is negative 12 plus or minus the square root of b squared , of 144 , that 's b squared minus 4 times a , which is negative 3 times c , which is 1 , all of that over 2 times a , over 2 times negative 3 . so all of that over negative 6 , this is going to be equal to negative 12 plus or minus the square root of -- what is this ? it 's a negative times a negative so they cancel out . so i have 144 plus 12 , so that is 156 , right ? 144 plus 12 , all of that over negative 6 . now , i suspect we can simplify this 156 . we could maybe bring some things out of the radical sign . so let 's attempt to do that . so let 's do a prime factorization of 156 . sometimes , this is the hardest part , simplifying the radical . so 156 is the same thing as 2 times 78 . 78 is the same thing as 2 times what ? that 's 2 times 39 . so the square root of 156 is equal to the square root of 2 times 2 times 39 or we could say that 's the square root of 2 times 2 times the square root of 39 . and this , obviously , is just going to be the square root of 4 or this is the square root of 2 times 2 is just 2 . 2 square roots of 39 , if i did that properly , let 's see , 4 times 39 . yeah , it looks like it 's right . so this up here will simplify to negative 12 plus or minus 2 times the square root of 39 , all of that over negative 6 . now we can divide the numerator and the denominator maybe by 2 . so this will be equal to negative 6 plus or minus the square root of 39 over negative 3 . or we could separate these two terms out . we could say this is equal to negative 6 over negative 3 plus or minus the square root of 39 over negative 3 . now , this is just a 2 right here , right ? these cancel out , 6 divided by 3 is 2 , so we get 2 . and now notice , if this is plus and we use this minus sign , the plus will become negative and the negative will become positive . but it still does n't matter , right ? we could say minus or plus , that 's the same thing as plus or minus the square root of 39 nine over 3 . i think that 's about as simple as we can get this answered . i want to make a very clear point of what i did that last step . i did not forget about this negative sign . i just said it does n't matter . it 's going to turn the positive into the negative ; it 's going to turn the negative into the positive . let me rewrite this . so this right here can be rewritten as 2 plus the square root of 39 over negative 3 or 2 minus the square root of 39 over negative 3 , right ? that 's what the plus or minus means , it could be this or that or both of them , really . now in this situation , this negative 3 will turn into 2 minus the square root of 39 over 3 , right ? i 'm just taking this negative out . here the negative and the negative will become a positive , and you get 2 plus the square root of 39 over 3 , right ? a negative times a negative is a positive . so once again , you have 2 plus or minus the square of 39 over 3 . 2 plus or minus the square root of 39 over 3 are solutions to this equation right there . let verify . i 'm just curious what the graph looks like . so let 's just look at it . let me clear this . where is the clear button ? so we have negative 3 three squared plus 12x plus 1 and let 's graph it . let 's see where it intersects the x-axis . it goes up there and then back down again . so 2 plus or minus the square , you see -- the square root of 39 is going to be a little bit more than 6 , right ? because 36 is 6 squared . so it 's going be a little bit more than 6 , so this is going to be a little bit more than 2 . a little bit more than 6 divided by 2 is a little bit more than 2 . so you 're going to get one value that 's a little bit more than 4 and then another value that should be a little bit less than 1 . and that looks like the case , you have 1 , 2 , 3 , 4 . you have a value that 's pretty close to 4 , and then you have another value that is a little bit -- it looks close to 0 but maybe a little bit less than that . so anyway , hopefully you found this application of the quadratic formula helpful .
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let 's rewrite the formula again , just in case we have n't had it memorized yet . x is going to be equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . i 'll supply this to another problem .
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when solving for x , why do you add -b plus or minus what has already been simplified and square rooted before diving by 2a ?
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in this video , i 'm going to expose you to what is maybe one of at least the top five most useful formulas in mathematics . and if you 've seen many of my videos , you know that i 'm not a big fan of memorizing things . but i will recommend you memorize it with the caveat that you also remember how to prove it , because i do n't want you to just remember things and not know where they came from . but with that said , let me show you what i 'm talking about : it 's the quadratic formula . and as you might guess , it is to solve for the roots , or the zeroes of quadratic equations . so let 's speak in very general terms and i 'll show you some examples . so let 's say i have an equation of the form ax squared plus bx plus c is equal to 0 . you should recognize this . this is a quadratic equation where a , b and c are -- well , a is the coefficient on the x squared term or the second degree term , b is the coefficient on the x term and then c , is , you could imagine , the coefficient on the x to the zero term , or it 's the constant term . now , given that you have a general quadratic equation like this , the quadratic formula tells us that the solutions to this equation are x is equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . and i know it seems crazy and convoluted and hard for you to memorize right now , but as you get a lot more practice you 'll see that it actually is a pretty reasonable formula to stick in your brain someplace . and you might say , gee , this is a wacky formula , where did it come from ? and in the next video i 'm going to show you where it came from . but i want you to get used to using it first . but it really just came from completing the square on this equation right there . if you complete the square here , you 're actually going to get this solution and that is the quadratic formula , right there . so let 's apply it to some problems . let 's start off with something that we could have factored just to verify that it 's giving us the same answer . so let 's say we have x squared plus 4x minus 21 is equal to 0 . so in this situation -- let me do that in a different color -- a is equal to 1 , right ? the coefficient on the x squared term is 1. b is equal to 4 , the coefficient on the x-term . and then c is equal to negative 21 , the constant term . and let 's just plug it in the formula , so what do we get ? we get x , this tells us that x is going to be equal to negative b . negative b is negative 4 -- i put the negative sign in front of that -- negative b plus or minus the square root of b squared . b squared is 16 , right ? 4 squared is 16 , minus 4 times a , which is 1 , times c , which is negative 21 . so we can put a 21 out there and that negative sign will cancel out just like that with that -- since this is the first time we 're doing it , let me not skip too many steps . so negative 21 , just so you can see how it fit in , and then all of that over 2a . a is 1 , so all of that over 2 . so what does this simplify , or hopefully it simplifies ? so we get x is equal to negative 4 plus or minus the square root of -- let 's see we have a negative times a negative , that 's going to give us a positive . and we had 16 plus , let 's see this is 6 , 4 times 1 is 4 times 21 is 84 . 16 plus 84 is 100 . that 's nice . that 's a nice perfect square . all of that over 2 , and so this is going to be equal to negative 4 plus or minus 10 over 2 . we could just divide both of these terms by 2 right now . so this is equal to negative 4 divided by 2 is negative 2 plus or minus 10 divided by 2 is 5 . so that tells us that x could be equal to negative 2 plus 5 , which is 3 , or x could be equal to negative 2 minus 5 , which is negative 7 . so the quadratic formula seems to have given us an answer for this . you can verify just by substituting back in that these do work , or you could even just try to factor this right here . you say what two numbers when you take their product , you get negative 21 and when you take their sum you get positive 4 ? so you 'd get x plus 7 times x minus 3 is equal to negative 21 . notice 7 times negative 3 is negative 21 , 7 minus 3 is positive 4 . you would get x plus -- sorry it 's not negative -- 21 is equal to 0 . there should be a 0 there . so you get x plus 7 is equal to 0 , or x minus 3 is equal to 0 . x could be equal to negative 7 or x could be equal to 3 . so it definitely gives us the same answer as factoring , so you might say , hey why bother with this crazy mess ? and the reason we want to bother with this crazy mess is it 'll also work for problems that are hard to factor . and let 's do a couple of those , let 's do some hard-to-factor problems right now . so let 's scroll down to get some fresh real estate . let 's rewrite the formula again , just in case we have n't had it memorized yet . x is going to be equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . i 'll supply this to another problem . let 's say we have the equation 3x squared plus 6x is equal to negative 10 . well , the first thing we want to do is get it in the form where all of our terms or on the left-hand side , so let 's add 10 to both sides of this equation . we get 3x squared plus the 6x plus 10 is equal to 0 . and now we can use a quadratic formula . so let 's apply it here . so a is equal to 3 . that is a , this is b and this right here is c. so the quadratic formula tells us the solutions to this equation . the roots of this quadratic function , i guess we could call it . x is going to be equal to negative b. b is 6 , so negative 6 plus or minus the square root of b squared . b is 6 , so we get 6 squared minus 4 times a , which is 3 times c , which is 10 . let 's stretch out the radical little bit , all of that over 2 times a , 2 times 3 . so we get x is equal to negative 6 plus or minus the square root of 36 minus -- this is interesting -- minus 4 times 3 times 10 . so this is minus -- 4 times 3 times 10 . so this is minus 120 . all of that over 6 . so this is interesting , you might already realize why it 's interesting . what is this going to simplify to ? 36 minus 120 is what ? that 's 84 . we make this into a 10 , this will become an 11 , this is a 4 . it is 84 , so this is going to be equal to negative 6 plus or minus the square root of -- but not positive 84 , that 's if it 's 120 minus 36 . we have 36 minus 120 . it 's going to be negative 84 all of that 6 . so you might say , gee , this is crazy . what a this silly quadratic formula you 're introducing me to , sal ? it 's worthless . it just gives me a square root of a negative number . it 's not giving me an answer . and the reason why it 's not giving you an answer , at least an answer that you might want , is because this will have no real solutions . in the future , we 're going to introduce something called an imaginary number , which is a square root of a negative number , and then we can actually express this in terms of those numbers . so this actually does have solutions , but they involve imaginary numbers . so this actually has no real solutions , we 're taking the square root of a negative number . so the b squared with the b squared minus 4ac , if this term right here is negative , then you 're not going to have any real solutions . and let 's verify that for ourselves . let 's get our graphic calculator out and let 's graph this equation right here . so , let 's get the graphs that y is equal to -- that 's what i had there before -- 3x squared plus 6x plus 10 . so that 's the equation and we 're going to see where it intersects the x-axis . where does it equal 0 ? so let me graph it . notice , this thing just comes down and then goes back up . its vertex is sitting here above the x-axis and it 's upward-opening . it never intersects the x-axis . so at no point will this expression , will this function , equal 0 . at no point will y equal 0 on this graph . so once again , the quadratic formula seems to be working . let 's do one more example , you can never see enough examples here . and i want to do ones that are , you know , maybe not so obvious to factor . so let 's say we get negative 3x squared plus 12x plus 1 is equal to 0 . now let 's try to do it just having the quadratic formula in our brain . so the x 's that satisfy this equation are going to be negative b . this is b so negative b is negative 12 plus or minus the square root of b squared , of 144 , that 's b squared minus 4 times a , which is negative 3 times c , which is 1 , all of that over 2 times a , over 2 times negative 3 . so all of that over negative 6 , this is going to be equal to negative 12 plus or minus the square root of -- what is this ? it 's a negative times a negative so they cancel out . so i have 144 plus 12 , so that is 156 , right ? 144 plus 12 , all of that over negative 6 . now , i suspect we can simplify this 156 . we could maybe bring some things out of the radical sign . so let 's attempt to do that . so let 's do a prime factorization of 156 . sometimes , this is the hardest part , simplifying the radical . so 156 is the same thing as 2 times 78 . 78 is the same thing as 2 times what ? that 's 2 times 39 . so the square root of 156 is equal to the square root of 2 times 2 times 39 or we could say that 's the square root of 2 times 2 times the square root of 39 . and this , obviously , is just going to be the square root of 4 or this is the square root of 2 times 2 is just 2 . 2 square roots of 39 , if i did that properly , let 's see , 4 times 39 . yeah , it looks like it 's right . so this up here will simplify to negative 12 plus or minus 2 times the square root of 39 , all of that over negative 6 . now we can divide the numerator and the denominator maybe by 2 . so this will be equal to negative 6 plus or minus the square root of 39 over negative 3 . or we could separate these two terms out . we could say this is equal to negative 6 over negative 3 plus or minus the square root of 39 over negative 3 . now , this is just a 2 right here , right ? these cancel out , 6 divided by 3 is 2 , so we get 2 . and now notice , if this is plus and we use this minus sign , the plus will become negative and the negative will become positive . but it still does n't matter , right ? we could say minus or plus , that 's the same thing as plus or minus the square root of 39 nine over 3 . i think that 's about as simple as we can get this answered . i want to make a very clear point of what i did that last step . i did not forget about this negative sign . i just said it does n't matter . it 's going to turn the positive into the negative ; it 's going to turn the negative into the positive . let me rewrite this . so this right here can be rewritten as 2 plus the square root of 39 over negative 3 or 2 minus the square root of 39 over negative 3 , right ? that 's what the plus or minus means , it could be this or that or both of them , really . now in this situation , this negative 3 will turn into 2 minus the square root of 39 over 3 , right ? i 'm just taking this negative out . here the negative and the negative will become a positive , and you get 2 plus the square root of 39 over 3 , right ? a negative times a negative is a positive . so once again , you have 2 plus or minus the square of 39 over 3 . 2 plus or minus the square root of 39 over 3 are solutions to this equation right there . let verify . i 'm just curious what the graph looks like . so let 's just look at it . let me clear this . where is the clear button ? so we have negative 3 three squared plus 12x plus 1 and let 's graph it . let 's see where it intersects the x-axis . it goes up there and then back down again . so 2 plus or minus the square , you see -- the square root of 39 is going to be a little bit more than 6 , right ? because 36 is 6 squared . so it 's going be a little bit more than 6 , so this is going to be a little bit more than 2 . a little bit more than 6 divided by 2 is a little bit more than 2 . so you 're going to get one value that 's a little bit more than 4 and then another value that should be a little bit less than 1 . and that looks like the case , you have 1 , 2 , 3 , 4 . you have a value that 's pretty close to 4 , and then you have another value that is a little bit -- it looks close to 0 but maybe a little bit less than that . so anyway , hopefully you found this application of the quadratic formula helpful .
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we get 3x squared plus the 6x plus 10 is equal to 0 . and now we can use a quadratic formula . so let 's apply it here .
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how do you get the quadratic formula from the original formula ?
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in this video , i 'm going to expose you to what is maybe one of at least the top five most useful formulas in mathematics . and if you 've seen many of my videos , you know that i 'm not a big fan of memorizing things . but i will recommend you memorize it with the caveat that you also remember how to prove it , because i do n't want you to just remember things and not know where they came from . but with that said , let me show you what i 'm talking about : it 's the quadratic formula . and as you might guess , it is to solve for the roots , or the zeroes of quadratic equations . so let 's speak in very general terms and i 'll show you some examples . so let 's say i have an equation of the form ax squared plus bx plus c is equal to 0 . you should recognize this . this is a quadratic equation where a , b and c are -- well , a is the coefficient on the x squared term or the second degree term , b is the coefficient on the x term and then c , is , you could imagine , the coefficient on the x to the zero term , or it 's the constant term . now , given that you have a general quadratic equation like this , the quadratic formula tells us that the solutions to this equation are x is equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . and i know it seems crazy and convoluted and hard for you to memorize right now , but as you get a lot more practice you 'll see that it actually is a pretty reasonable formula to stick in your brain someplace . and you might say , gee , this is a wacky formula , where did it come from ? and in the next video i 'm going to show you where it came from . but i want you to get used to using it first . but it really just came from completing the square on this equation right there . if you complete the square here , you 're actually going to get this solution and that is the quadratic formula , right there . so let 's apply it to some problems . let 's start off with something that we could have factored just to verify that it 's giving us the same answer . so let 's say we have x squared plus 4x minus 21 is equal to 0 . so in this situation -- let me do that in a different color -- a is equal to 1 , right ? the coefficient on the x squared term is 1. b is equal to 4 , the coefficient on the x-term . and then c is equal to negative 21 , the constant term . and let 's just plug it in the formula , so what do we get ? we get x , this tells us that x is going to be equal to negative b . negative b is negative 4 -- i put the negative sign in front of that -- negative b plus or minus the square root of b squared . b squared is 16 , right ? 4 squared is 16 , minus 4 times a , which is 1 , times c , which is negative 21 . so we can put a 21 out there and that negative sign will cancel out just like that with that -- since this is the first time we 're doing it , let me not skip too many steps . so negative 21 , just so you can see how it fit in , and then all of that over 2a . a is 1 , so all of that over 2 . so what does this simplify , or hopefully it simplifies ? so we get x is equal to negative 4 plus or minus the square root of -- let 's see we have a negative times a negative , that 's going to give us a positive . and we had 16 plus , let 's see this is 6 , 4 times 1 is 4 times 21 is 84 . 16 plus 84 is 100 . that 's nice . that 's a nice perfect square . all of that over 2 , and so this is going to be equal to negative 4 plus or minus 10 over 2 . we could just divide both of these terms by 2 right now . so this is equal to negative 4 divided by 2 is negative 2 plus or minus 10 divided by 2 is 5 . so that tells us that x could be equal to negative 2 plus 5 , which is 3 , or x could be equal to negative 2 minus 5 , which is negative 7 . so the quadratic formula seems to have given us an answer for this . you can verify just by substituting back in that these do work , or you could even just try to factor this right here . you say what two numbers when you take their product , you get negative 21 and when you take their sum you get positive 4 ? so you 'd get x plus 7 times x minus 3 is equal to negative 21 . notice 7 times negative 3 is negative 21 , 7 minus 3 is positive 4 . you would get x plus -- sorry it 's not negative -- 21 is equal to 0 . there should be a 0 there . so you get x plus 7 is equal to 0 , or x minus 3 is equal to 0 . x could be equal to negative 7 or x could be equal to 3 . so it definitely gives us the same answer as factoring , so you might say , hey why bother with this crazy mess ? and the reason we want to bother with this crazy mess is it 'll also work for problems that are hard to factor . and let 's do a couple of those , let 's do some hard-to-factor problems right now . so let 's scroll down to get some fresh real estate . let 's rewrite the formula again , just in case we have n't had it memorized yet . x is going to be equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . i 'll supply this to another problem . let 's say we have the equation 3x squared plus 6x is equal to negative 10 . well , the first thing we want to do is get it in the form where all of our terms or on the left-hand side , so let 's add 10 to both sides of this equation . we get 3x squared plus the 6x plus 10 is equal to 0 . and now we can use a quadratic formula . so let 's apply it here . so a is equal to 3 . that is a , this is b and this right here is c. so the quadratic formula tells us the solutions to this equation . the roots of this quadratic function , i guess we could call it . x is going to be equal to negative b. b is 6 , so negative 6 plus or minus the square root of b squared . b is 6 , so we get 6 squared minus 4 times a , which is 3 times c , which is 10 . let 's stretch out the radical little bit , all of that over 2 times a , 2 times 3 . so we get x is equal to negative 6 plus or minus the square root of 36 minus -- this is interesting -- minus 4 times 3 times 10 . so this is minus -- 4 times 3 times 10 . so this is minus 120 . all of that over 6 . so this is interesting , you might already realize why it 's interesting . what is this going to simplify to ? 36 minus 120 is what ? that 's 84 . we make this into a 10 , this will become an 11 , this is a 4 . it is 84 , so this is going to be equal to negative 6 plus or minus the square root of -- but not positive 84 , that 's if it 's 120 minus 36 . we have 36 minus 120 . it 's going to be negative 84 all of that 6 . so you might say , gee , this is crazy . what a this silly quadratic formula you 're introducing me to , sal ? it 's worthless . it just gives me a square root of a negative number . it 's not giving me an answer . and the reason why it 's not giving you an answer , at least an answer that you might want , is because this will have no real solutions . in the future , we 're going to introduce something called an imaginary number , which is a square root of a negative number , and then we can actually express this in terms of those numbers . so this actually does have solutions , but they involve imaginary numbers . so this actually has no real solutions , we 're taking the square root of a negative number . so the b squared with the b squared minus 4ac , if this term right here is negative , then you 're not going to have any real solutions . and let 's verify that for ourselves . let 's get our graphic calculator out and let 's graph this equation right here . so , let 's get the graphs that y is equal to -- that 's what i had there before -- 3x squared plus 6x plus 10 . so that 's the equation and we 're going to see where it intersects the x-axis . where does it equal 0 ? so let me graph it . notice , this thing just comes down and then goes back up . its vertex is sitting here above the x-axis and it 's upward-opening . it never intersects the x-axis . so at no point will this expression , will this function , equal 0 . at no point will y equal 0 on this graph . so once again , the quadratic formula seems to be working . let 's do one more example , you can never see enough examples here . and i want to do ones that are , you know , maybe not so obvious to factor . so let 's say we get negative 3x squared plus 12x plus 1 is equal to 0 . now let 's try to do it just having the quadratic formula in our brain . so the x 's that satisfy this equation are going to be negative b . this is b so negative b is negative 12 plus or minus the square root of b squared , of 144 , that 's b squared minus 4 times a , which is negative 3 times c , which is 1 , all of that over 2 times a , over 2 times negative 3 . so all of that over negative 6 , this is going to be equal to negative 12 plus or minus the square root of -- what is this ? it 's a negative times a negative so they cancel out . so i have 144 plus 12 , so that is 156 , right ? 144 plus 12 , all of that over negative 6 . now , i suspect we can simplify this 156 . we could maybe bring some things out of the radical sign . so let 's attempt to do that . so let 's do a prime factorization of 156 . sometimes , this is the hardest part , simplifying the radical . so 156 is the same thing as 2 times 78 . 78 is the same thing as 2 times what ? that 's 2 times 39 . so the square root of 156 is equal to the square root of 2 times 2 times 39 or we could say that 's the square root of 2 times 2 times the square root of 39 . and this , obviously , is just going to be the square root of 4 or this is the square root of 2 times 2 is just 2 . 2 square roots of 39 , if i did that properly , let 's see , 4 times 39 . yeah , it looks like it 's right . so this up here will simplify to negative 12 plus or minus 2 times the square root of 39 , all of that over negative 6 . now we can divide the numerator and the denominator maybe by 2 . so this will be equal to negative 6 plus or minus the square root of 39 over negative 3 . or we could separate these two terms out . we could say this is equal to negative 6 over negative 3 plus or minus the square root of 39 over negative 3 . now , this is just a 2 right here , right ? these cancel out , 6 divided by 3 is 2 , so we get 2 . and now notice , if this is plus and we use this minus sign , the plus will become negative and the negative will become positive . but it still does n't matter , right ? we could say minus or plus , that 's the same thing as plus or minus the square root of 39 nine over 3 . i think that 's about as simple as we can get this answered . i want to make a very clear point of what i did that last step . i did not forget about this negative sign . i just said it does n't matter . it 's going to turn the positive into the negative ; it 's going to turn the negative into the positive . let me rewrite this . so this right here can be rewritten as 2 plus the square root of 39 over negative 3 or 2 minus the square root of 39 over negative 3 , right ? that 's what the plus or minus means , it could be this or that or both of them , really . now in this situation , this negative 3 will turn into 2 minus the square root of 39 over 3 , right ? i 'm just taking this negative out . here the negative and the negative will become a positive , and you get 2 plus the square root of 39 over 3 , right ? a negative times a negative is a positive . so once again , you have 2 plus or minus the square of 39 over 3 . 2 plus or minus the square root of 39 over 3 are solutions to this equation right there . let verify . i 'm just curious what the graph looks like . so let 's just look at it . let me clear this . where is the clear button ? so we have negative 3 three squared plus 12x plus 1 and let 's graph it . let 's see where it intersects the x-axis . it goes up there and then back down again . so 2 plus or minus the square , you see -- the square root of 39 is going to be a little bit more than 6 , right ? because 36 is 6 squared . so it 's going be a little bit more than 6 , so this is going to be a little bit more than 2 . a little bit more than 6 divided by 2 is a little bit more than 2 . so you 're going to get one value that 's a little bit more than 4 and then another value that should be a little bit less than 1 . and that looks like the case , you have 1 , 2 , 3 , 4 . you have a value that 's pretty close to 4 , and then you have another value that is a little bit -- it looks close to 0 but maybe a little bit less than that . so anyway , hopefully you found this application of the quadratic formula helpful .
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we get 3x squared plus the 6x plus 10 is equal to 0 . and now we can use a quadratic formula . so let 's apply it here .
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what is factoring , why do we use it , and how is it useful ?
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in this video , i 'm going to expose you to what is maybe one of at least the top five most useful formulas in mathematics . and if you 've seen many of my videos , you know that i 'm not a big fan of memorizing things . but i will recommend you memorize it with the caveat that you also remember how to prove it , because i do n't want you to just remember things and not know where they came from . but with that said , let me show you what i 'm talking about : it 's the quadratic formula . and as you might guess , it is to solve for the roots , or the zeroes of quadratic equations . so let 's speak in very general terms and i 'll show you some examples . so let 's say i have an equation of the form ax squared plus bx plus c is equal to 0 . you should recognize this . this is a quadratic equation where a , b and c are -- well , a is the coefficient on the x squared term or the second degree term , b is the coefficient on the x term and then c , is , you could imagine , the coefficient on the x to the zero term , or it 's the constant term . now , given that you have a general quadratic equation like this , the quadratic formula tells us that the solutions to this equation are x is equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . and i know it seems crazy and convoluted and hard for you to memorize right now , but as you get a lot more practice you 'll see that it actually is a pretty reasonable formula to stick in your brain someplace . and you might say , gee , this is a wacky formula , where did it come from ? and in the next video i 'm going to show you where it came from . but i want you to get used to using it first . but it really just came from completing the square on this equation right there . if you complete the square here , you 're actually going to get this solution and that is the quadratic formula , right there . so let 's apply it to some problems . let 's start off with something that we could have factored just to verify that it 's giving us the same answer . so let 's say we have x squared plus 4x minus 21 is equal to 0 . so in this situation -- let me do that in a different color -- a is equal to 1 , right ? the coefficient on the x squared term is 1. b is equal to 4 , the coefficient on the x-term . and then c is equal to negative 21 , the constant term . and let 's just plug it in the formula , so what do we get ? we get x , this tells us that x is going to be equal to negative b . negative b is negative 4 -- i put the negative sign in front of that -- negative b plus or minus the square root of b squared . b squared is 16 , right ? 4 squared is 16 , minus 4 times a , which is 1 , times c , which is negative 21 . so we can put a 21 out there and that negative sign will cancel out just like that with that -- since this is the first time we 're doing it , let me not skip too many steps . so negative 21 , just so you can see how it fit in , and then all of that over 2a . a is 1 , so all of that over 2 . so what does this simplify , or hopefully it simplifies ? so we get x is equal to negative 4 plus or minus the square root of -- let 's see we have a negative times a negative , that 's going to give us a positive . and we had 16 plus , let 's see this is 6 , 4 times 1 is 4 times 21 is 84 . 16 plus 84 is 100 . that 's nice . that 's a nice perfect square . all of that over 2 , and so this is going to be equal to negative 4 plus or minus 10 over 2 . we could just divide both of these terms by 2 right now . so this is equal to negative 4 divided by 2 is negative 2 plus or minus 10 divided by 2 is 5 . so that tells us that x could be equal to negative 2 plus 5 , which is 3 , or x could be equal to negative 2 minus 5 , which is negative 7 . so the quadratic formula seems to have given us an answer for this . you can verify just by substituting back in that these do work , or you could even just try to factor this right here . you say what two numbers when you take their product , you get negative 21 and when you take their sum you get positive 4 ? so you 'd get x plus 7 times x minus 3 is equal to negative 21 . notice 7 times negative 3 is negative 21 , 7 minus 3 is positive 4 . you would get x plus -- sorry it 's not negative -- 21 is equal to 0 . there should be a 0 there . so you get x plus 7 is equal to 0 , or x minus 3 is equal to 0 . x could be equal to negative 7 or x could be equal to 3 . so it definitely gives us the same answer as factoring , so you might say , hey why bother with this crazy mess ? and the reason we want to bother with this crazy mess is it 'll also work for problems that are hard to factor . and let 's do a couple of those , let 's do some hard-to-factor problems right now . so let 's scroll down to get some fresh real estate . let 's rewrite the formula again , just in case we have n't had it memorized yet . x is going to be equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . i 'll supply this to another problem . let 's say we have the equation 3x squared plus 6x is equal to negative 10 . well , the first thing we want to do is get it in the form where all of our terms or on the left-hand side , so let 's add 10 to both sides of this equation . we get 3x squared plus the 6x plus 10 is equal to 0 . and now we can use a quadratic formula . so let 's apply it here . so a is equal to 3 . that is a , this is b and this right here is c. so the quadratic formula tells us the solutions to this equation . the roots of this quadratic function , i guess we could call it . x is going to be equal to negative b. b is 6 , so negative 6 plus or minus the square root of b squared . b is 6 , so we get 6 squared minus 4 times a , which is 3 times c , which is 10 . let 's stretch out the radical little bit , all of that over 2 times a , 2 times 3 . so we get x is equal to negative 6 plus or minus the square root of 36 minus -- this is interesting -- minus 4 times 3 times 10 . so this is minus -- 4 times 3 times 10 . so this is minus 120 . all of that over 6 . so this is interesting , you might already realize why it 's interesting . what is this going to simplify to ? 36 minus 120 is what ? that 's 84 . we make this into a 10 , this will become an 11 , this is a 4 . it is 84 , so this is going to be equal to negative 6 plus or minus the square root of -- but not positive 84 , that 's if it 's 120 minus 36 . we have 36 minus 120 . it 's going to be negative 84 all of that 6 . so you might say , gee , this is crazy . what a this silly quadratic formula you 're introducing me to , sal ? it 's worthless . it just gives me a square root of a negative number . it 's not giving me an answer . and the reason why it 's not giving you an answer , at least an answer that you might want , is because this will have no real solutions . in the future , we 're going to introduce something called an imaginary number , which is a square root of a negative number , and then we can actually express this in terms of those numbers . so this actually does have solutions , but they involve imaginary numbers . so this actually has no real solutions , we 're taking the square root of a negative number . so the b squared with the b squared minus 4ac , if this term right here is negative , then you 're not going to have any real solutions . and let 's verify that for ourselves . let 's get our graphic calculator out and let 's graph this equation right here . so , let 's get the graphs that y is equal to -- that 's what i had there before -- 3x squared plus 6x plus 10 . so that 's the equation and we 're going to see where it intersects the x-axis . where does it equal 0 ? so let me graph it . notice , this thing just comes down and then goes back up . its vertex is sitting here above the x-axis and it 's upward-opening . it never intersects the x-axis . so at no point will this expression , will this function , equal 0 . at no point will y equal 0 on this graph . so once again , the quadratic formula seems to be working . let 's do one more example , you can never see enough examples here . and i want to do ones that are , you know , maybe not so obvious to factor . so let 's say we get negative 3x squared plus 12x plus 1 is equal to 0 . now let 's try to do it just having the quadratic formula in our brain . so the x 's that satisfy this equation are going to be negative b . this is b so negative b is negative 12 plus or minus the square root of b squared , of 144 , that 's b squared minus 4 times a , which is negative 3 times c , which is 1 , all of that over 2 times a , over 2 times negative 3 . so all of that over negative 6 , this is going to be equal to negative 12 plus or minus the square root of -- what is this ? it 's a negative times a negative so they cancel out . so i have 144 plus 12 , so that is 156 , right ? 144 plus 12 , all of that over negative 6 . now , i suspect we can simplify this 156 . we could maybe bring some things out of the radical sign . so let 's attempt to do that . so let 's do a prime factorization of 156 . sometimes , this is the hardest part , simplifying the radical . so 156 is the same thing as 2 times 78 . 78 is the same thing as 2 times what ? that 's 2 times 39 . so the square root of 156 is equal to the square root of 2 times 2 times 39 or we could say that 's the square root of 2 times 2 times the square root of 39 . and this , obviously , is just going to be the square root of 4 or this is the square root of 2 times 2 is just 2 . 2 square roots of 39 , if i did that properly , let 's see , 4 times 39 . yeah , it looks like it 's right . so this up here will simplify to negative 12 plus or minus 2 times the square root of 39 , all of that over negative 6 . now we can divide the numerator and the denominator maybe by 2 . so this will be equal to negative 6 plus or minus the square root of 39 over negative 3 . or we could separate these two terms out . we could say this is equal to negative 6 over negative 3 plus or minus the square root of 39 over negative 3 . now , this is just a 2 right here , right ? these cancel out , 6 divided by 3 is 2 , so we get 2 . and now notice , if this is plus and we use this minus sign , the plus will become negative and the negative will become positive . but it still does n't matter , right ? we could say minus or plus , that 's the same thing as plus or minus the square root of 39 nine over 3 . i think that 's about as simple as we can get this answered . i want to make a very clear point of what i did that last step . i did not forget about this negative sign . i just said it does n't matter . it 's going to turn the positive into the negative ; it 's going to turn the negative into the positive . let me rewrite this . so this right here can be rewritten as 2 plus the square root of 39 over negative 3 or 2 minus the square root of 39 over negative 3 , right ? that 's what the plus or minus means , it could be this or that or both of them , really . now in this situation , this negative 3 will turn into 2 minus the square root of 39 over 3 , right ? i 'm just taking this negative out . here the negative and the negative will become a positive , and you get 2 plus the square root of 39 over 3 , right ? a negative times a negative is a positive . so once again , you have 2 plus or minus the square of 39 over 3 . 2 plus or minus the square root of 39 over 3 are solutions to this equation right there . let verify . i 'm just curious what the graph looks like . so let 's just look at it . let me clear this . where is the clear button ? so we have negative 3 three squared plus 12x plus 1 and let 's graph it . let 's see where it intersects the x-axis . it goes up there and then back down again . so 2 plus or minus the square , you see -- the square root of 39 is going to be a little bit more than 6 , right ? because 36 is 6 squared . so it 's going be a little bit more than 6 , so this is going to be a little bit more than 2 . a little bit more than 6 divided by 2 is a little bit more than 2 . so you 're going to get one value that 's a little bit more than 4 and then another value that should be a little bit less than 1 . and that looks like the case , you have 1 , 2 , 3 , 4 . you have a value that 's pretty close to 4 , and then you have another value that is a little bit -- it looks close to 0 but maybe a little bit less than that . so anyway , hopefully you found this application of the quadratic formula helpful .
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b squared is 16 , right ? 4 squared is 16 , minus 4 times a , which is 1 , times c , which is negative 21 . so we can put a 21 out there and that negative sign will cancel out just like that with that -- since this is the first time we 're doing it , let me not skip too many steps .
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why did sal write -4 for multiplying a and c ?
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in this video , i 'm going to expose you to what is maybe one of at least the top five most useful formulas in mathematics . and if you 've seen many of my videos , you know that i 'm not a big fan of memorizing things . but i will recommend you memorize it with the caveat that you also remember how to prove it , because i do n't want you to just remember things and not know where they came from . but with that said , let me show you what i 'm talking about : it 's the quadratic formula . and as you might guess , it is to solve for the roots , or the zeroes of quadratic equations . so let 's speak in very general terms and i 'll show you some examples . so let 's say i have an equation of the form ax squared plus bx plus c is equal to 0 . you should recognize this . this is a quadratic equation where a , b and c are -- well , a is the coefficient on the x squared term or the second degree term , b is the coefficient on the x term and then c , is , you could imagine , the coefficient on the x to the zero term , or it 's the constant term . now , given that you have a general quadratic equation like this , the quadratic formula tells us that the solutions to this equation are x is equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . and i know it seems crazy and convoluted and hard for you to memorize right now , but as you get a lot more practice you 'll see that it actually is a pretty reasonable formula to stick in your brain someplace . and you might say , gee , this is a wacky formula , where did it come from ? and in the next video i 'm going to show you where it came from . but i want you to get used to using it first . but it really just came from completing the square on this equation right there . if you complete the square here , you 're actually going to get this solution and that is the quadratic formula , right there . so let 's apply it to some problems . let 's start off with something that we could have factored just to verify that it 's giving us the same answer . so let 's say we have x squared plus 4x minus 21 is equal to 0 . so in this situation -- let me do that in a different color -- a is equal to 1 , right ? the coefficient on the x squared term is 1. b is equal to 4 , the coefficient on the x-term . and then c is equal to negative 21 , the constant term . and let 's just plug it in the formula , so what do we get ? we get x , this tells us that x is going to be equal to negative b . negative b is negative 4 -- i put the negative sign in front of that -- negative b plus or minus the square root of b squared . b squared is 16 , right ? 4 squared is 16 , minus 4 times a , which is 1 , times c , which is negative 21 . so we can put a 21 out there and that negative sign will cancel out just like that with that -- since this is the first time we 're doing it , let me not skip too many steps . so negative 21 , just so you can see how it fit in , and then all of that over 2a . a is 1 , so all of that over 2 . so what does this simplify , or hopefully it simplifies ? so we get x is equal to negative 4 plus or minus the square root of -- let 's see we have a negative times a negative , that 's going to give us a positive . and we had 16 plus , let 's see this is 6 , 4 times 1 is 4 times 21 is 84 . 16 plus 84 is 100 . that 's nice . that 's a nice perfect square . all of that over 2 , and so this is going to be equal to negative 4 plus or minus 10 over 2 . we could just divide both of these terms by 2 right now . so this is equal to negative 4 divided by 2 is negative 2 plus or minus 10 divided by 2 is 5 . so that tells us that x could be equal to negative 2 plus 5 , which is 3 , or x could be equal to negative 2 minus 5 , which is negative 7 . so the quadratic formula seems to have given us an answer for this . you can verify just by substituting back in that these do work , or you could even just try to factor this right here . you say what two numbers when you take their product , you get negative 21 and when you take their sum you get positive 4 ? so you 'd get x plus 7 times x minus 3 is equal to negative 21 . notice 7 times negative 3 is negative 21 , 7 minus 3 is positive 4 . you would get x plus -- sorry it 's not negative -- 21 is equal to 0 . there should be a 0 there . so you get x plus 7 is equal to 0 , or x minus 3 is equal to 0 . x could be equal to negative 7 or x could be equal to 3 . so it definitely gives us the same answer as factoring , so you might say , hey why bother with this crazy mess ? and the reason we want to bother with this crazy mess is it 'll also work for problems that are hard to factor . and let 's do a couple of those , let 's do some hard-to-factor problems right now . so let 's scroll down to get some fresh real estate . let 's rewrite the formula again , just in case we have n't had it memorized yet . x is going to be equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . i 'll supply this to another problem . let 's say we have the equation 3x squared plus 6x is equal to negative 10 . well , the first thing we want to do is get it in the form where all of our terms or on the left-hand side , so let 's add 10 to both sides of this equation . we get 3x squared plus the 6x plus 10 is equal to 0 . and now we can use a quadratic formula . so let 's apply it here . so a is equal to 3 . that is a , this is b and this right here is c. so the quadratic formula tells us the solutions to this equation . the roots of this quadratic function , i guess we could call it . x is going to be equal to negative b. b is 6 , so negative 6 plus or minus the square root of b squared . b is 6 , so we get 6 squared minus 4 times a , which is 3 times c , which is 10 . let 's stretch out the radical little bit , all of that over 2 times a , 2 times 3 . so we get x is equal to negative 6 plus or minus the square root of 36 minus -- this is interesting -- minus 4 times 3 times 10 . so this is minus -- 4 times 3 times 10 . so this is minus 120 . all of that over 6 . so this is interesting , you might already realize why it 's interesting . what is this going to simplify to ? 36 minus 120 is what ? that 's 84 . we make this into a 10 , this will become an 11 , this is a 4 . it is 84 , so this is going to be equal to negative 6 plus or minus the square root of -- but not positive 84 , that 's if it 's 120 minus 36 . we have 36 minus 120 . it 's going to be negative 84 all of that 6 . so you might say , gee , this is crazy . what a this silly quadratic formula you 're introducing me to , sal ? it 's worthless . it just gives me a square root of a negative number . it 's not giving me an answer . and the reason why it 's not giving you an answer , at least an answer that you might want , is because this will have no real solutions . in the future , we 're going to introduce something called an imaginary number , which is a square root of a negative number , and then we can actually express this in terms of those numbers . so this actually does have solutions , but they involve imaginary numbers . so this actually has no real solutions , we 're taking the square root of a negative number . so the b squared with the b squared minus 4ac , if this term right here is negative , then you 're not going to have any real solutions . and let 's verify that for ourselves . let 's get our graphic calculator out and let 's graph this equation right here . so , let 's get the graphs that y is equal to -- that 's what i had there before -- 3x squared plus 6x plus 10 . so that 's the equation and we 're going to see where it intersects the x-axis . where does it equal 0 ? so let me graph it . notice , this thing just comes down and then goes back up . its vertex is sitting here above the x-axis and it 's upward-opening . it never intersects the x-axis . so at no point will this expression , will this function , equal 0 . at no point will y equal 0 on this graph . so once again , the quadratic formula seems to be working . let 's do one more example , you can never see enough examples here . and i want to do ones that are , you know , maybe not so obvious to factor . so let 's say we get negative 3x squared plus 12x plus 1 is equal to 0 . now let 's try to do it just having the quadratic formula in our brain . so the x 's that satisfy this equation are going to be negative b . this is b so negative b is negative 12 plus or minus the square root of b squared , of 144 , that 's b squared minus 4 times a , which is negative 3 times c , which is 1 , all of that over 2 times a , over 2 times negative 3 . so all of that over negative 6 , this is going to be equal to negative 12 plus or minus the square root of -- what is this ? it 's a negative times a negative so they cancel out . so i have 144 plus 12 , so that is 156 , right ? 144 plus 12 , all of that over negative 6 . now , i suspect we can simplify this 156 . we could maybe bring some things out of the radical sign . so let 's attempt to do that . so let 's do a prime factorization of 156 . sometimes , this is the hardest part , simplifying the radical . so 156 is the same thing as 2 times 78 . 78 is the same thing as 2 times what ? that 's 2 times 39 . so the square root of 156 is equal to the square root of 2 times 2 times 39 or we could say that 's the square root of 2 times 2 times the square root of 39 . and this , obviously , is just going to be the square root of 4 or this is the square root of 2 times 2 is just 2 . 2 square roots of 39 , if i did that properly , let 's see , 4 times 39 . yeah , it looks like it 's right . so this up here will simplify to negative 12 plus or minus 2 times the square root of 39 , all of that over negative 6 . now we can divide the numerator and the denominator maybe by 2 . so this will be equal to negative 6 plus or minus the square root of 39 over negative 3 . or we could separate these two terms out . we could say this is equal to negative 6 over negative 3 plus or minus the square root of 39 over negative 3 . now , this is just a 2 right here , right ? these cancel out , 6 divided by 3 is 2 , so we get 2 . and now notice , if this is plus and we use this minus sign , the plus will become negative and the negative will become positive . but it still does n't matter , right ? we could say minus or plus , that 's the same thing as plus or minus the square root of 39 nine over 3 . i think that 's about as simple as we can get this answered . i want to make a very clear point of what i did that last step . i did not forget about this negative sign . i just said it does n't matter . it 's going to turn the positive into the negative ; it 's going to turn the negative into the positive . let me rewrite this . so this right here can be rewritten as 2 plus the square root of 39 over negative 3 or 2 minus the square root of 39 over negative 3 , right ? that 's what the plus or minus means , it could be this or that or both of them , really . now in this situation , this negative 3 will turn into 2 minus the square root of 39 over 3 , right ? i 'm just taking this negative out . here the negative and the negative will become a positive , and you get 2 plus the square root of 39 over 3 , right ? a negative times a negative is a positive . so once again , you have 2 plus or minus the square of 39 over 3 . 2 plus or minus the square root of 39 over 3 are solutions to this equation right there . let verify . i 'm just curious what the graph looks like . so let 's just look at it . let me clear this . where is the clear button ? so we have negative 3 three squared plus 12x plus 1 and let 's graph it . let 's see where it intersects the x-axis . it goes up there and then back down again . so 2 plus or minus the square , you see -- the square root of 39 is going to be a little bit more than 6 , right ? because 36 is 6 squared . so it 's going be a little bit more than 6 , so this is going to be a little bit more than 2 . a little bit more than 6 divided by 2 is a little bit more than 2 . so you 're going to get one value that 's a little bit more than 4 and then another value that should be a little bit less than 1 . and that looks like the case , you have 1 , 2 , 3 , 4 . you have a value that 's pretty close to 4 , and then you have another value that is a little bit -- it looks close to 0 but maybe a little bit less than that . so anyway , hopefully you found this application of the quadratic formula helpful .
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so negative 21 , just so you can see how it fit in , and then all of that over 2a . a is 1 , so all of that over 2 . so what does this simplify , or hopefully it simplifies ?
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will a always be 1 ?
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in this video , i 'm going to expose you to what is maybe one of at least the top five most useful formulas in mathematics . and if you 've seen many of my videos , you know that i 'm not a big fan of memorizing things . but i will recommend you memorize it with the caveat that you also remember how to prove it , because i do n't want you to just remember things and not know where they came from . but with that said , let me show you what i 'm talking about : it 's the quadratic formula . and as you might guess , it is to solve for the roots , or the zeroes of quadratic equations . so let 's speak in very general terms and i 'll show you some examples . so let 's say i have an equation of the form ax squared plus bx plus c is equal to 0 . you should recognize this . this is a quadratic equation where a , b and c are -- well , a is the coefficient on the x squared term or the second degree term , b is the coefficient on the x term and then c , is , you could imagine , the coefficient on the x to the zero term , or it 's the constant term . now , given that you have a general quadratic equation like this , the quadratic formula tells us that the solutions to this equation are x is equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . and i know it seems crazy and convoluted and hard for you to memorize right now , but as you get a lot more practice you 'll see that it actually is a pretty reasonable formula to stick in your brain someplace . and you might say , gee , this is a wacky formula , where did it come from ? and in the next video i 'm going to show you where it came from . but i want you to get used to using it first . but it really just came from completing the square on this equation right there . if you complete the square here , you 're actually going to get this solution and that is the quadratic formula , right there . so let 's apply it to some problems . let 's start off with something that we could have factored just to verify that it 's giving us the same answer . so let 's say we have x squared plus 4x minus 21 is equal to 0 . so in this situation -- let me do that in a different color -- a is equal to 1 , right ? the coefficient on the x squared term is 1. b is equal to 4 , the coefficient on the x-term . and then c is equal to negative 21 , the constant term . and let 's just plug it in the formula , so what do we get ? we get x , this tells us that x is going to be equal to negative b . negative b is negative 4 -- i put the negative sign in front of that -- negative b plus or minus the square root of b squared . b squared is 16 , right ? 4 squared is 16 , minus 4 times a , which is 1 , times c , which is negative 21 . so we can put a 21 out there and that negative sign will cancel out just like that with that -- since this is the first time we 're doing it , let me not skip too many steps . so negative 21 , just so you can see how it fit in , and then all of that over 2a . a is 1 , so all of that over 2 . so what does this simplify , or hopefully it simplifies ? so we get x is equal to negative 4 plus or minus the square root of -- let 's see we have a negative times a negative , that 's going to give us a positive . and we had 16 plus , let 's see this is 6 , 4 times 1 is 4 times 21 is 84 . 16 plus 84 is 100 . that 's nice . that 's a nice perfect square . all of that over 2 , and so this is going to be equal to negative 4 plus or minus 10 over 2 . we could just divide both of these terms by 2 right now . so this is equal to negative 4 divided by 2 is negative 2 plus or minus 10 divided by 2 is 5 . so that tells us that x could be equal to negative 2 plus 5 , which is 3 , or x could be equal to negative 2 minus 5 , which is negative 7 . so the quadratic formula seems to have given us an answer for this . you can verify just by substituting back in that these do work , or you could even just try to factor this right here . you say what two numbers when you take their product , you get negative 21 and when you take their sum you get positive 4 ? so you 'd get x plus 7 times x minus 3 is equal to negative 21 . notice 7 times negative 3 is negative 21 , 7 minus 3 is positive 4 . you would get x plus -- sorry it 's not negative -- 21 is equal to 0 . there should be a 0 there . so you get x plus 7 is equal to 0 , or x minus 3 is equal to 0 . x could be equal to negative 7 or x could be equal to 3 . so it definitely gives us the same answer as factoring , so you might say , hey why bother with this crazy mess ? and the reason we want to bother with this crazy mess is it 'll also work for problems that are hard to factor . and let 's do a couple of those , let 's do some hard-to-factor problems right now . so let 's scroll down to get some fresh real estate . let 's rewrite the formula again , just in case we have n't had it memorized yet . x is going to be equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . i 'll supply this to another problem . let 's say we have the equation 3x squared plus 6x is equal to negative 10 . well , the first thing we want to do is get it in the form where all of our terms or on the left-hand side , so let 's add 10 to both sides of this equation . we get 3x squared plus the 6x plus 10 is equal to 0 . and now we can use a quadratic formula . so let 's apply it here . so a is equal to 3 . that is a , this is b and this right here is c. so the quadratic formula tells us the solutions to this equation . the roots of this quadratic function , i guess we could call it . x is going to be equal to negative b. b is 6 , so negative 6 plus or minus the square root of b squared . b is 6 , so we get 6 squared minus 4 times a , which is 3 times c , which is 10 . let 's stretch out the radical little bit , all of that over 2 times a , 2 times 3 . so we get x is equal to negative 6 plus or minus the square root of 36 minus -- this is interesting -- minus 4 times 3 times 10 . so this is minus -- 4 times 3 times 10 . so this is minus 120 . all of that over 6 . so this is interesting , you might already realize why it 's interesting . what is this going to simplify to ? 36 minus 120 is what ? that 's 84 . we make this into a 10 , this will become an 11 , this is a 4 . it is 84 , so this is going to be equal to negative 6 plus or minus the square root of -- but not positive 84 , that 's if it 's 120 minus 36 . we have 36 minus 120 . it 's going to be negative 84 all of that 6 . so you might say , gee , this is crazy . what a this silly quadratic formula you 're introducing me to , sal ? it 's worthless . it just gives me a square root of a negative number . it 's not giving me an answer . and the reason why it 's not giving you an answer , at least an answer that you might want , is because this will have no real solutions . in the future , we 're going to introduce something called an imaginary number , which is a square root of a negative number , and then we can actually express this in terms of those numbers . so this actually does have solutions , but they involve imaginary numbers . so this actually has no real solutions , we 're taking the square root of a negative number . so the b squared with the b squared minus 4ac , if this term right here is negative , then you 're not going to have any real solutions . and let 's verify that for ourselves . let 's get our graphic calculator out and let 's graph this equation right here . so , let 's get the graphs that y is equal to -- that 's what i had there before -- 3x squared plus 6x plus 10 . so that 's the equation and we 're going to see where it intersects the x-axis . where does it equal 0 ? so let me graph it . notice , this thing just comes down and then goes back up . its vertex is sitting here above the x-axis and it 's upward-opening . it never intersects the x-axis . so at no point will this expression , will this function , equal 0 . at no point will y equal 0 on this graph . so once again , the quadratic formula seems to be working . let 's do one more example , you can never see enough examples here . and i want to do ones that are , you know , maybe not so obvious to factor . so let 's say we get negative 3x squared plus 12x plus 1 is equal to 0 . now let 's try to do it just having the quadratic formula in our brain . so the x 's that satisfy this equation are going to be negative b . this is b so negative b is negative 12 plus or minus the square root of b squared , of 144 , that 's b squared minus 4 times a , which is negative 3 times c , which is 1 , all of that over 2 times a , over 2 times negative 3 . so all of that over negative 6 , this is going to be equal to negative 12 plus or minus the square root of -- what is this ? it 's a negative times a negative so they cancel out . so i have 144 plus 12 , so that is 156 , right ? 144 plus 12 , all of that over negative 6 . now , i suspect we can simplify this 156 . we could maybe bring some things out of the radical sign . so let 's attempt to do that . so let 's do a prime factorization of 156 . sometimes , this is the hardest part , simplifying the radical . so 156 is the same thing as 2 times 78 . 78 is the same thing as 2 times what ? that 's 2 times 39 . so the square root of 156 is equal to the square root of 2 times 2 times 39 or we could say that 's the square root of 2 times 2 times the square root of 39 . and this , obviously , is just going to be the square root of 4 or this is the square root of 2 times 2 is just 2 . 2 square roots of 39 , if i did that properly , let 's see , 4 times 39 . yeah , it looks like it 's right . so this up here will simplify to negative 12 plus or minus 2 times the square root of 39 , all of that over negative 6 . now we can divide the numerator and the denominator maybe by 2 . so this will be equal to negative 6 plus or minus the square root of 39 over negative 3 . or we could separate these two terms out . we could say this is equal to negative 6 over negative 3 plus or minus the square root of 39 over negative 3 . now , this is just a 2 right here , right ? these cancel out , 6 divided by 3 is 2 , so we get 2 . and now notice , if this is plus and we use this minus sign , the plus will become negative and the negative will become positive . but it still does n't matter , right ? we could say minus or plus , that 's the same thing as plus or minus the square root of 39 nine over 3 . i think that 's about as simple as we can get this answered . i want to make a very clear point of what i did that last step . i did not forget about this negative sign . i just said it does n't matter . it 's going to turn the positive into the negative ; it 's going to turn the negative into the positive . let me rewrite this . so this right here can be rewritten as 2 plus the square root of 39 over negative 3 or 2 minus the square root of 39 over negative 3 , right ? that 's what the plus or minus means , it could be this or that or both of them , really . now in this situation , this negative 3 will turn into 2 minus the square root of 39 over 3 , right ? i 'm just taking this negative out . here the negative and the negative will become a positive , and you get 2 plus the square root of 39 over 3 , right ? a negative times a negative is a positive . so once again , you have 2 plus or minus the square of 39 over 3 . 2 plus or minus the square root of 39 over 3 are solutions to this equation right there . let verify . i 'm just curious what the graph looks like . so let 's just look at it . let me clear this . where is the clear button ? so we have negative 3 three squared plus 12x plus 1 and let 's graph it . let 's see where it intersects the x-axis . it goes up there and then back down again . so 2 plus or minus the square , you see -- the square root of 39 is going to be a little bit more than 6 , right ? because 36 is 6 squared . so it 's going be a little bit more than 6 , so this is going to be a little bit more than 2 . a little bit more than 6 divided by 2 is a little bit more than 2 . so you 're going to get one value that 's a little bit more than 4 and then another value that should be a little bit less than 1 . and that looks like the case , you have 1 , 2 , 3 , 4 . you have a value that 's pretty close to 4 , and then you have another value that is a little bit -- it looks close to 0 but maybe a little bit less than that . so anyway , hopefully you found this application of the quadratic formula helpful .
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you say what two numbers when you take their product , you get negative 21 and when you take their sum you get positive 4 ? so you 'd get x plus 7 times x minus 3 is equal to negative 21 . notice 7 times negative 3 is negative 21 , 7 minus 3 is positive 4 .
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because a 1 is in front of the x ?
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in this video , i 'm going to expose you to what is maybe one of at least the top five most useful formulas in mathematics . and if you 've seen many of my videos , you know that i 'm not a big fan of memorizing things . but i will recommend you memorize it with the caveat that you also remember how to prove it , because i do n't want you to just remember things and not know where they came from . but with that said , let me show you what i 'm talking about : it 's the quadratic formula . and as you might guess , it is to solve for the roots , or the zeroes of quadratic equations . so let 's speak in very general terms and i 'll show you some examples . so let 's say i have an equation of the form ax squared plus bx plus c is equal to 0 . you should recognize this . this is a quadratic equation where a , b and c are -- well , a is the coefficient on the x squared term or the second degree term , b is the coefficient on the x term and then c , is , you could imagine , the coefficient on the x to the zero term , or it 's the constant term . now , given that you have a general quadratic equation like this , the quadratic formula tells us that the solutions to this equation are x is equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . and i know it seems crazy and convoluted and hard for you to memorize right now , but as you get a lot more practice you 'll see that it actually is a pretty reasonable formula to stick in your brain someplace . and you might say , gee , this is a wacky formula , where did it come from ? and in the next video i 'm going to show you where it came from . but i want you to get used to using it first . but it really just came from completing the square on this equation right there . if you complete the square here , you 're actually going to get this solution and that is the quadratic formula , right there . so let 's apply it to some problems . let 's start off with something that we could have factored just to verify that it 's giving us the same answer . so let 's say we have x squared plus 4x minus 21 is equal to 0 . so in this situation -- let me do that in a different color -- a is equal to 1 , right ? the coefficient on the x squared term is 1. b is equal to 4 , the coefficient on the x-term . and then c is equal to negative 21 , the constant term . and let 's just plug it in the formula , so what do we get ? we get x , this tells us that x is going to be equal to negative b . negative b is negative 4 -- i put the negative sign in front of that -- negative b plus or minus the square root of b squared . b squared is 16 , right ? 4 squared is 16 , minus 4 times a , which is 1 , times c , which is negative 21 . so we can put a 21 out there and that negative sign will cancel out just like that with that -- since this is the first time we 're doing it , let me not skip too many steps . so negative 21 , just so you can see how it fit in , and then all of that over 2a . a is 1 , so all of that over 2 . so what does this simplify , or hopefully it simplifies ? so we get x is equal to negative 4 plus or minus the square root of -- let 's see we have a negative times a negative , that 's going to give us a positive . and we had 16 plus , let 's see this is 6 , 4 times 1 is 4 times 21 is 84 . 16 plus 84 is 100 . that 's nice . that 's a nice perfect square . all of that over 2 , and so this is going to be equal to negative 4 plus or minus 10 over 2 . we could just divide both of these terms by 2 right now . so this is equal to negative 4 divided by 2 is negative 2 plus or minus 10 divided by 2 is 5 . so that tells us that x could be equal to negative 2 plus 5 , which is 3 , or x could be equal to negative 2 minus 5 , which is negative 7 . so the quadratic formula seems to have given us an answer for this . you can verify just by substituting back in that these do work , or you could even just try to factor this right here . you say what two numbers when you take their product , you get negative 21 and when you take their sum you get positive 4 ? so you 'd get x plus 7 times x minus 3 is equal to negative 21 . notice 7 times negative 3 is negative 21 , 7 minus 3 is positive 4 . you would get x plus -- sorry it 's not negative -- 21 is equal to 0 . there should be a 0 there . so you get x plus 7 is equal to 0 , or x minus 3 is equal to 0 . x could be equal to negative 7 or x could be equal to 3 . so it definitely gives us the same answer as factoring , so you might say , hey why bother with this crazy mess ? and the reason we want to bother with this crazy mess is it 'll also work for problems that are hard to factor . and let 's do a couple of those , let 's do some hard-to-factor problems right now . so let 's scroll down to get some fresh real estate . let 's rewrite the formula again , just in case we have n't had it memorized yet . x is going to be equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . i 'll supply this to another problem . let 's say we have the equation 3x squared plus 6x is equal to negative 10 . well , the first thing we want to do is get it in the form where all of our terms or on the left-hand side , so let 's add 10 to both sides of this equation . we get 3x squared plus the 6x plus 10 is equal to 0 . and now we can use a quadratic formula . so let 's apply it here . so a is equal to 3 . that is a , this is b and this right here is c. so the quadratic formula tells us the solutions to this equation . the roots of this quadratic function , i guess we could call it . x is going to be equal to negative b. b is 6 , so negative 6 plus or minus the square root of b squared . b is 6 , so we get 6 squared minus 4 times a , which is 3 times c , which is 10 . let 's stretch out the radical little bit , all of that over 2 times a , 2 times 3 . so we get x is equal to negative 6 plus or minus the square root of 36 minus -- this is interesting -- minus 4 times 3 times 10 . so this is minus -- 4 times 3 times 10 . so this is minus 120 . all of that over 6 . so this is interesting , you might already realize why it 's interesting . what is this going to simplify to ? 36 minus 120 is what ? that 's 84 . we make this into a 10 , this will become an 11 , this is a 4 . it is 84 , so this is going to be equal to negative 6 plus or minus the square root of -- but not positive 84 , that 's if it 's 120 minus 36 . we have 36 minus 120 . it 's going to be negative 84 all of that 6 . so you might say , gee , this is crazy . what a this silly quadratic formula you 're introducing me to , sal ? it 's worthless . it just gives me a square root of a negative number . it 's not giving me an answer . and the reason why it 's not giving you an answer , at least an answer that you might want , is because this will have no real solutions . in the future , we 're going to introduce something called an imaginary number , which is a square root of a negative number , and then we can actually express this in terms of those numbers . so this actually does have solutions , but they involve imaginary numbers . so this actually has no real solutions , we 're taking the square root of a negative number . so the b squared with the b squared minus 4ac , if this term right here is negative , then you 're not going to have any real solutions . and let 's verify that for ourselves . let 's get our graphic calculator out and let 's graph this equation right here . so , let 's get the graphs that y is equal to -- that 's what i had there before -- 3x squared plus 6x plus 10 . so that 's the equation and we 're going to see where it intersects the x-axis . where does it equal 0 ? so let me graph it . notice , this thing just comes down and then goes back up . its vertex is sitting here above the x-axis and it 's upward-opening . it never intersects the x-axis . so at no point will this expression , will this function , equal 0 . at no point will y equal 0 on this graph . so once again , the quadratic formula seems to be working . let 's do one more example , you can never see enough examples here . and i want to do ones that are , you know , maybe not so obvious to factor . so let 's say we get negative 3x squared plus 12x plus 1 is equal to 0 . now let 's try to do it just having the quadratic formula in our brain . so the x 's that satisfy this equation are going to be negative b . this is b so negative b is negative 12 plus or minus the square root of b squared , of 144 , that 's b squared minus 4 times a , which is negative 3 times c , which is 1 , all of that over 2 times a , over 2 times negative 3 . so all of that over negative 6 , this is going to be equal to negative 12 plus or minus the square root of -- what is this ? it 's a negative times a negative so they cancel out . so i have 144 plus 12 , so that is 156 , right ? 144 plus 12 , all of that over negative 6 . now , i suspect we can simplify this 156 . we could maybe bring some things out of the radical sign . so let 's attempt to do that . so let 's do a prime factorization of 156 . sometimes , this is the hardest part , simplifying the radical . so 156 is the same thing as 2 times 78 . 78 is the same thing as 2 times what ? that 's 2 times 39 . so the square root of 156 is equal to the square root of 2 times 2 times 39 or we could say that 's the square root of 2 times 2 times the square root of 39 . and this , obviously , is just going to be the square root of 4 or this is the square root of 2 times 2 is just 2 . 2 square roots of 39 , if i did that properly , let 's see , 4 times 39 . yeah , it looks like it 's right . so this up here will simplify to negative 12 plus or minus 2 times the square root of 39 , all of that over negative 6 . now we can divide the numerator and the denominator maybe by 2 . so this will be equal to negative 6 plus or minus the square root of 39 over negative 3 . or we could separate these two terms out . we could say this is equal to negative 6 over negative 3 plus or minus the square root of 39 over negative 3 . now , this is just a 2 right here , right ? these cancel out , 6 divided by 3 is 2 , so we get 2 . and now notice , if this is plus and we use this minus sign , the plus will become negative and the negative will become positive . but it still does n't matter , right ? we could say minus or plus , that 's the same thing as plus or minus the square root of 39 nine over 3 . i think that 's about as simple as we can get this answered . i want to make a very clear point of what i did that last step . i did not forget about this negative sign . i just said it does n't matter . it 's going to turn the positive into the negative ; it 's going to turn the negative into the positive . let me rewrite this . so this right here can be rewritten as 2 plus the square root of 39 over negative 3 or 2 minus the square root of 39 over negative 3 , right ? that 's what the plus or minus means , it could be this or that or both of them , really . now in this situation , this negative 3 will turn into 2 minus the square root of 39 over 3 , right ? i 'm just taking this negative out . here the negative and the negative will become a positive , and you get 2 plus the square root of 39 over 3 , right ? a negative times a negative is a positive . so once again , you have 2 plus or minus the square of 39 over 3 . 2 plus or minus the square root of 39 over 3 are solutions to this equation right there . let verify . i 'm just curious what the graph looks like . so let 's just look at it . let me clear this . where is the clear button ? so we have negative 3 three squared plus 12x plus 1 and let 's graph it . let 's see where it intersects the x-axis . it goes up there and then back down again . so 2 plus or minus the square , you see -- the square root of 39 is going to be a little bit more than 6 , right ? because 36 is 6 squared . so it 's going be a little bit more than 6 , so this is going to be a little bit more than 2 . a little bit more than 6 divided by 2 is a little bit more than 2 . so you 're going to get one value that 's a little bit more than 4 and then another value that should be a little bit less than 1 . and that looks like the case , you have 1 , 2 , 3 , 4 . you have a value that 's pretty close to 4 , and then you have another value that is a little bit -- it looks close to 0 but maybe a little bit less than that . so anyway , hopefully you found this application of the quadratic formula helpful .
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that 's 2 times 39 . so the square root of 156 is equal to the square root of 2 times 2 times 39 or we could say that 's the square root of 2 times 2 times the square root of 39 . and this , obviously , is just going to be the square root of 4 or this is the square root of 2 times 2 is just 2 .
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what is a integral root ?
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in this video , i 'm going to expose you to what is maybe one of at least the top five most useful formulas in mathematics . and if you 've seen many of my videos , you know that i 'm not a big fan of memorizing things . but i will recommend you memorize it with the caveat that you also remember how to prove it , because i do n't want you to just remember things and not know where they came from . but with that said , let me show you what i 'm talking about : it 's the quadratic formula . and as you might guess , it is to solve for the roots , or the zeroes of quadratic equations . so let 's speak in very general terms and i 'll show you some examples . so let 's say i have an equation of the form ax squared plus bx plus c is equal to 0 . you should recognize this . this is a quadratic equation where a , b and c are -- well , a is the coefficient on the x squared term or the second degree term , b is the coefficient on the x term and then c , is , you could imagine , the coefficient on the x to the zero term , or it 's the constant term . now , given that you have a general quadratic equation like this , the quadratic formula tells us that the solutions to this equation are x is equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . and i know it seems crazy and convoluted and hard for you to memorize right now , but as you get a lot more practice you 'll see that it actually is a pretty reasonable formula to stick in your brain someplace . and you might say , gee , this is a wacky formula , where did it come from ? and in the next video i 'm going to show you where it came from . but i want you to get used to using it first . but it really just came from completing the square on this equation right there . if you complete the square here , you 're actually going to get this solution and that is the quadratic formula , right there . so let 's apply it to some problems . let 's start off with something that we could have factored just to verify that it 's giving us the same answer . so let 's say we have x squared plus 4x minus 21 is equal to 0 . so in this situation -- let me do that in a different color -- a is equal to 1 , right ? the coefficient on the x squared term is 1. b is equal to 4 , the coefficient on the x-term . and then c is equal to negative 21 , the constant term . and let 's just plug it in the formula , so what do we get ? we get x , this tells us that x is going to be equal to negative b . negative b is negative 4 -- i put the negative sign in front of that -- negative b plus or minus the square root of b squared . b squared is 16 , right ? 4 squared is 16 , minus 4 times a , which is 1 , times c , which is negative 21 . so we can put a 21 out there and that negative sign will cancel out just like that with that -- since this is the first time we 're doing it , let me not skip too many steps . so negative 21 , just so you can see how it fit in , and then all of that over 2a . a is 1 , so all of that over 2 . so what does this simplify , or hopefully it simplifies ? so we get x is equal to negative 4 plus or minus the square root of -- let 's see we have a negative times a negative , that 's going to give us a positive . and we had 16 plus , let 's see this is 6 , 4 times 1 is 4 times 21 is 84 . 16 plus 84 is 100 . that 's nice . that 's a nice perfect square . all of that over 2 , and so this is going to be equal to negative 4 plus or minus 10 over 2 . we could just divide both of these terms by 2 right now . so this is equal to negative 4 divided by 2 is negative 2 plus or minus 10 divided by 2 is 5 . so that tells us that x could be equal to negative 2 plus 5 , which is 3 , or x could be equal to negative 2 minus 5 , which is negative 7 . so the quadratic formula seems to have given us an answer for this . you can verify just by substituting back in that these do work , or you could even just try to factor this right here . you say what two numbers when you take their product , you get negative 21 and when you take their sum you get positive 4 ? so you 'd get x plus 7 times x minus 3 is equal to negative 21 . notice 7 times negative 3 is negative 21 , 7 minus 3 is positive 4 . you would get x plus -- sorry it 's not negative -- 21 is equal to 0 . there should be a 0 there . so you get x plus 7 is equal to 0 , or x minus 3 is equal to 0 . x could be equal to negative 7 or x could be equal to 3 . so it definitely gives us the same answer as factoring , so you might say , hey why bother with this crazy mess ? and the reason we want to bother with this crazy mess is it 'll also work for problems that are hard to factor . and let 's do a couple of those , let 's do some hard-to-factor problems right now . so let 's scroll down to get some fresh real estate . let 's rewrite the formula again , just in case we have n't had it memorized yet . x is going to be equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . i 'll supply this to another problem . let 's say we have the equation 3x squared plus 6x is equal to negative 10 . well , the first thing we want to do is get it in the form where all of our terms or on the left-hand side , so let 's add 10 to both sides of this equation . we get 3x squared plus the 6x plus 10 is equal to 0 . and now we can use a quadratic formula . so let 's apply it here . so a is equal to 3 . that is a , this is b and this right here is c. so the quadratic formula tells us the solutions to this equation . the roots of this quadratic function , i guess we could call it . x is going to be equal to negative b. b is 6 , so negative 6 plus or minus the square root of b squared . b is 6 , so we get 6 squared minus 4 times a , which is 3 times c , which is 10 . let 's stretch out the radical little bit , all of that over 2 times a , 2 times 3 . so we get x is equal to negative 6 plus or minus the square root of 36 minus -- this is interesting -- minus 4 times 3 times 10 . so this is minus -- 4 times 3 times 10 . so this is minus 120 . all of that over 6 . so this is interesting , you might already realize why it 's interesting . what is this going to simplify to ? 36 minus 120 is what ? that 's 84 . we make this into a 10 , this will become an 11 , this is a 4 . it is 84 , so this is going to be equal to negative 6 plus or minus the square root of -- but not positive 84 , that 's if it 's 120 minus 36 . we have 36 minus 120 . it 's going to be negative 84 all of that 6 . so you might say , gee , this is crazy . what a this silly quadratic formula you 're introducing me to , sal ? it 's worthless . it just gives me a square root of a negative number . it 's not giving me an answer . and the reason why it 's not giving you an answer , at least an answer that you might want , is because this will have no real solutions . in the future , we 're going to introduce something called an imaginary number , which is a square root of a negative number , and then we can actually express this in terms of those numbers . so this actually does have solutions , but they involve imaginary numbers . so this actually has no real solutions , we 're taking the square root of a negative number . so the b squared with the b squared minus 4ac , if this term right here is negative , then you 're not going to have any real solutions . and let 's verify that for ourselves . let 's get our graphic calculator out and let 's graph this equation right here . so , let 's get the graphs that y is equal to -- that 's what i had there before -- 3x squared plus 6x plus 10 . so that 's the equation and we 're going to see where it intersects the x-axis . where does it equal 0 ? so let me graph it . notice , this thing just comes down and then goes back up . its vertex is sitting here above the x-axis and it 's upward-opening . it never intersects the x-axis . so at no point will this expression , will this function , equal 0 . at no point will y equal 0 on this graph . so once again , the quadratic formula seems to be working . let 's do one more example , you can never see enough examples here . and i want to do ones that are , you know , maybe not so obvious to factor . so let 's say we get negative 3x squared plus 12x plus 1 is equal to 0 . now let 's try to do it just having the quadratic formula in our brain . so the x 's that satisfy this equation are going to be negative b . this is b so negative b is negative 12 plus or minus the square root of b squared , of 144 , that 's b squared minus 4 times a , which is negative 3 times c , which is 1 , all of that over 2 times a , over 2 times negative 3 . so all of that over negative 6 , this is going to be equal to negative 12 plus or minus the square root of -- what is this ? it 's a negative times a negative so they cancel out . so i have 144 plus 12 , so that is 156 , right ? 144 plus 12 , all of that over negative 6 . now , i suspect we can simplify this 156 . we could maybe bring some things out of the radical sign . so let 's attempt to do that . so let 's do a prime factorization of 156 . sometimes , this is the hardest part , simplifying the radical . so 156 is the same thing as 2 times 78 . 78 is the same thing as 2 times what ? that 's 2 times 39 . so the square root of 156 is equal to the square root of 2 times 2 times 39 or we could say that 's the square root of 2 times 2 times the square root of 39 . and this , obviously , is just going to be the square root of 4 or this is the square root of 2 times 2 is just 2 . 2 square roots of 39 , if i did that properly , let 's see , 4 times 39 . yeah , it looks like it 's right . so this up here will simplify to negative 12 plus or minus 2 times the square root of 39 , all of that over negative 6 . now we can divide the numerator and the denominator maybe by 2 . so this will be equal to negative 6 plus or minus the square root of 39 over negative 3 . or we could separate these two terms out . we could say this is equal to negative 6 over negative 3 plus or minus the square root of 39 over negative 3 . now , this is just a 2 right here , right ? these cancel out , 6 divided by 3 is 2 , so we get 2 . and now notice , if this is plus and we use this minus sign , the plus will become negative and the negative will become positive . but it still does n't matter , right ? we could say minus or plus , that 's the same thing as plus or minus the square root of 39 nine over 3 . i think that 's about as simple as we can get this answered . i want to make a very clear point of what i did that last step . i did not forget about this negative sign . i just said it does n't matter . it 's going to turn the positive into the negative ; it 's going to turn the negative into the positive . let me rewrite this . so this right here can be rewritten as 2 plus the square root of 39 over negative 3 or 2 minus the square root of 39 over negative 3 , right ? that 's what the plus or minus means , it could be this or that or both of them , really . now in this situation , this negative 3 will turn into 2 minus the square root of 39 over 3 , right ? i 'm just taking this negative out . here the negative and the negative will become a positive , and you get 2 plus the square root of 39 over 3 , right ? a negative times a negative is a positive . so once again , you have 2 plus or minus the square of 39 over 3 . 2 plus or minus the square root of 39 over 3 are solutions to this equation right there . let verify . i 'm just curious what the graph looks like . so let 's just look at it . let me clear this . where is the clear button ? so we have negative 3 three squared plus 12x plus 1 and let 's graph it . let 's see where it intersects the x-axis . it goes up there and then back down again . so 2 plus or minus the square , you see -- the square root of 39 is going to be a little bit more than 6 , right ? because 36 is 6 squared . so it 's going be a little bit more than 6 , so this is going to be a little bit more than 2 . a little bit more than 6 divided by 2 is a little bit more than 2 . so you 're going to get one value that 's a little bit more than 4 and then another value that should be a little bit less than 1 . and that looks like the case , you have 1 , 2 , 3 , 4 . you have a value that 's pretty close to 4 , and then you have another value that is a little bit -- it looks close to 0 but maybe a little bit less than that . so anyway , hopefully you found this application of the quadratic formula helpful .
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its vertex is sitting here above the x-axis and it 's upward-opening . it never intersects the x-axis . so at no point will this expression , will this function , equal 0 .
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in problems in which the quadratic formula solves as a solution which never intersects the x or y axis will the solution intersect the imaginary axes in the imaginary graph ?
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in this video , i 'm going to expose you to what is maybe one of at least the top five most useful formulas in mathematics . and if you 've seen many of my videos , you know that i 'm not a big fan of memorizing things . but i will recommend you memorize it with the caveat that you also remember how to prove it , because i do n't want you to just remember things and not know where they came from . but with that said , let me show you what i 'm talking about : it 's the quadratic formula . and as you might guess , it is to solve for the roots , or the zeroes of quadratic equations . so let 's speak in very general terms and i 'll show you some examples . so let 's say i have an equation of the form ax squared plus bx plus c is equal to 0 . you should recognize this . this is a quadratic equation where a , b and c are -- well , a is the coefficient on the x squared term or the second degree term , b is the coefficient on the x term and then c , is , you could imagine , the coefficient on the x to the zero term , or it 's the constant term . now , given that you have a general quadratic equation like this , the quadratic formula tells us that the solutions to this equation are x is equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . and i know it seems crazy and convoluted and hard for you to memorize right now , but as you get a lot more practice you 'll see that it actually is a pretty reasonable formula to stick in your brain someplace . and you might say , gee , this is a wacky formula , where did it come from ? and in the next video i 'm going to show you where it came from . but i want you to get used to using it first . but it really just came from completing the square on this equation right there . if you complete the square here , you 're actually going to get this solution and that is the quadratic formula , right there . so let 's apply it to some problems . let 's start off with something that we could have factored just to verify that it 's giving us the same answer . so let 's say we have x squared plus 4x minus 21 is equal to 0 . so in this situation -- let me do that in a different color -- a is equal to 1 , right ? the coefficient on the x squared term is 1. b is equal to 4 , the coefficient on the x-term . and then c is equal to negative 21 , the constant term . and let 's just plug it in the formula , so what do we get ? we get x , this tells us that x is going to be equal to negative b . negative b is negative 4 -- i put the negative sign in front of that -- negative b plus or minus the square root of b squared . b squared is 16 , right ? 4 squared is 16 , minus 4 times a , which is 1 , times c , which is negative 21 . so we can put a 21 out there and that negative sign will cancel out just like that with that -- since this is the first time we 're doing it , let me not skip too many steps . so negative 21 , just so you can see how it fit in , and then all of that over 2a . a is 1 , so all of that over 2 . so what does this simplify , or hopefully it simplifies ? so we get x is equal to negative 4 plus or minus the square root of -- let 's see we have a negative times a negative , that 's going to give us a positive . and we had 16 plus , let 's see this is 6 , 4 times 1 is 4 times 21 is 84 . 16 plus 84 is 100 . that 's nice . that 's a nice perfect square . all of that over 2 , and so this is going to be equal to negative 4 plus or minus 10 over 2 . we could just divide both of these terms by 2 right now . so this is equal to negative 4 divided by 2 is negative 2 plus or minus 10 divided by 2 is 5 . so that tells us that x could be equal to negative 2 plus 5 , which is 3 , or x could be equal to negative 2 minus 5 , which is negative 7 . so the quadratic formula seems to have given us an answer for this . you can verify just by substituting back in that these do work , or you could even just try to factor this right here . you say what two numbers when you take their product , you get negative 21 and when you take their sum you get positive 4 ? so you 'd get x plus 7 times x minus 3 is equal to negative 21 . notice 7 times negative 3 is negative 21 , 7 minus 3 is positive 4 . you would get x plus -- sorry it 's not negative -- 21 is equal to 0 . there should be a 0 there . so you get x plus 7 is equal to 0 , or x minus 3 is equal to 0 . x could be equal to negative 7 or x could be equal to 3 . so it definitely gives us the same answer as factoring , so you might say , hey why bother with this crazy mess ? and the reason we want to bother with this crazy mess is it 'll also work for problems that are hard to factor . and let 's do a couple of those , let 's do some hard-to-factor problems right now . so let 's scroll down to get some fresh real estate . let 's rewrite the formula again , just in case we have n't had it memorized yet . x is going to be equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . i 'll supply this to another problem . let 's say we have the equation 3x squared plus 6x is equal to negative 10 . well , the first thing we want to do is get it in the form where all of our terms or on the left-hand side , so let 's add 10 to both sides of this equation . we get 3x squared plus the 6x plus 10 is equal to 0 . and now we can use a quadratic formula . so let 's apply it here . so a is equal to 3 . that is a , this is b and this right here is c. so the quadratic formula tells us the solutions to this equation . the roots of this quadratic function , i guess we could call it . x is going to be equal to negative b. b is 6 , so negative 6 plus or minus the square root of b squared . b is 6 , so we get 6 squared minus 4 times a , which is 3 times c , which is 10 . let 's stretch out the radical little bit , all of that over 2 times a , 2 times 3 . so we get x is equal to negative 6 plus or minus the square root of 36 minus -- this is interesting -- minus 4 times 3 times 10 . so this is minus -- 4 times 3 times 10 . so this is minus 120 . all of that over 6 . so this is interesting , you might already realize why it 's interesting . what is this going to simplify to ? 36 minus 120 is what ? that 's 84 . we make this into a 10 , this will become an 11 , this is a 4 . it is 84 , so this is going to be equal to negative 6 plus or minus the square root of -- but not positive 84 , that 's if it 's 120 minus 36 . we have 36 minus 120 . it 's going to be negative 84 all of that 6 . so you might say , gee , this is crazy . what a this silly quadratic formula you 're introducing me to , sal ? it 's worthless . it just gives me a square root of a negative number . it 's not giving me an answer . and the reason why it 's not giving you an answer , at least an answer that you might want , is because this will have no real solutions . in the future , we 're going to introduce something called an imaginary number , which is a square root of a negative number , and then we can actually express this in terms of those numbers . so this actually does have solutions , but they involve imaginary numbers . so this actually has no real solutions , we 're taking the square root of a negative number . so the b squared with the b squared minus 4ac , if this term right here is negative , then you 're not going to have any real solutions . and let 's verify that for ourselves . let 's get our graphic calculator out and let 's graph this equation right here . so , let 's get the graphs that y is equal to -- that 's what i had there before -- 3x squared plus 6x plus 10 . so that 's the equation and we 're going to see where it intersects the x-axis . where does it equal 0 ? so let me graph it . notice , this thing just comes down and then goes back up . its vertex is sitting here above the x-axis and it 's upward-opening . it never intersects the x-axis . so at no point will this expression , will this function , equal 0 . at no point will y equal 0 on this graph . so once again , the quadratic formula seems to be working . let 's do one more example , you can never see enough examples here . and i want to do ones that are , you know , maybe not so obvious to factor . so let 's say we get negative 3x squared plus 12x plus 1 is equal to 0 . now let 's try to do it just having the quadratic formula in our brain . so the x 's that satisfy this equation are going to be negative b . this is b so negative b is negative 12 plus or minus the square root of b squared , of 144 , that 's b squared minus 4 times a , which is negative 3 times c , which is 1 , all of that over 2 times a , over 2 times negative 3 . so all of that over negative 6 , this is going to be equal to negative 12 plus or minus the square root of -- what is this ? it 's a negative times a negative so they cancel out . so i have 144 plus 12 , so that is 156 , right ? 144 plus 12 , all of that over negative 6 . now , i suspect we can simplify this 156 . we could maybe bring some things out of the radical sign . so let 's attempt to do that . so let 's do a prime factorization of 156 . sometimes , this is the hardest part , simplifying the radical . so 156 is the same thing as 2 times 78 . 78 is the same thing as 2 times what ? that 's 2 times 39 . so the square root of 156 is equal to the square root of 2 times 2 times 39 or we could say that 's the square root of 2 times 2 times the square root of 39 . and this , obviously , is just going to be the square root of 4 or this is the square root of 2 times 2 is just 2 . 2 square roots of 39 , if i did that properly , let 's see , 4 times 39 . yeah , it looks like it 's right . so this up here will simplify to negative 12 plus or minus 2 times the square root of 39 , all of that over negative 6 . now we can divide the numerator and the denominator maybe by 2 . so this will be equal to negative 6 plus or minus the square root of 39 over negative 3 . or we could separate these two terms out . we could say this is equal to negative 6 over negative 3 plus or minus the square root of 39 over negative 3 . now , this is just a 2 right here , right ? these cancel out , 6 divided by 3 is 2 , so we get 2 . and now notice , if this is plus and we use this minus sign , the plus will become negative and the negative will become positive . but it still does n't matter , right ? we could say minus or plus , that 's the same thing as plus or minus the square root of 39 nine over 3 . i think that 's about as simple as we can get this answered . i want to make a very clear point of what i did that last step . i did not forget about this negative sign . i just said it does n't matter . it 's going to turn the positive into the negative ; it 's going to turn the negative into the positive . let me rewrite this . so this right here can be rewritten as 2 plus the square root of 39 over negative 3 or 2 minus the square root of 39 over negative 3 , right ? that 's what the plus or minus means , it could be this or that or both of them , really . now in this situation , this negative 3 will turn into 2 minus the square root of 39 over 3 , right ? i 'm just taking this negative out . here the negative and the negative will become a positive , and you get 2 plus the square root of 39 over 3 , right ? a negative times a negative is a positive . so once again , you have 2 plus or minus the square of 39 over 3 . 2 plus or minus the square root of 39 over 3 are solutions to this equation right there . let verify . i 'm just curious what the graph looks like . so let 's just look at it . let me clear this . where is the clear button ? so we have negative 3 three squared plus 12x plus 1 and let 's graph it . let 's see where it intersects the x-axis . it goes up there and then back down again . so 2 plus or minus the square , you see -- the square root of 39 is going to be a little bit more than 6 , right ? because 36 is 6 squared . so it 's going be a little bit more than 6 , so this is going to be a little bit more than 2 . a little bit more than 6 divided by 2 is a little bit more than 2 . so you 're going to get one value that 's a little bit more than 4 and then another value that should be a little bit less than 1 . and that looks like the case , you have 1 , 2 , 3 , 4 . you have a value that 's pretty close to 4 , and then you have another value that is a little bit -- it looks close to 0 but maybe a little bit less than that . so anyway , hopefully you found this application of the quadratic formula helpful .
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this is a quadratic equation where a , b and c are -- well , a is the coefficient on the x squared term or the second degree term , b is the coefficient on the x term and then c , is , you could imagine , the coefficient on the x to the zero term , or it 's the constant term . now , given that you have a general quadratic equation like this , the quadratic formula tells us that the solutions to this equation are x is equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . and i know it seems crazy and convoluted and hard for you to memorize right now , but as you get a lot more practice you 'll see that it actually is a pretty reasonable formula to stick in your brain someplace .
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i remember my 7th grade math teacher said `` five tomato '' to remember 5280 feet in a mile , so is there a trick like that which can be used on the quadratic equation ?
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in this video , i 'm going to expose you to what is maybe one of at least the top five most useful formulas in mathematics . and if you 've seen many of my videos , you know that i 'm not a big fan of memorizing things . but i will recommend you memorize it with the caveat that you also remember how to prove it , because i do n't want you to just remember things and not know where they came from . but with that said , let me show you what i 'm talking about : it 's the quadratic formula . and as you might guess , it is to solve for the roots , or the zeroes of quadratic equations . so let 's speak in very general terms and i 'll show you some examples . so let 's say i have an equation of the form ax squared plus bx plus c is equal to 0 . you should recognize this . this is a quadratic equation where a , b and c are -- well , a is the coefficient on the x squared term or the second degree term , b is the coefficient on the x term and then c , is , you could imagine , the coefficient on the x to the zero term , or it 's the constant term . now , given that you have a general quadratic equation like this , the quadratic formula tells us that the solutions to this equation are x is equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . and i know it seems crazy and convoluted and hard for you to memorize right now , but as you get a lot more practice you 'll see that it actually is a pretty reasonable formula to stick in your brain someplace . and you might say , gee , this is a wacky formula , where did it come from ? and in the next video i 'm going to show you where it came from . but i want you to get used to using it first . but it really just came from completing the square on this equation right there . if you complete the square here , you 're actually going to get this solution and that is the quadratic formula , right there . so let 's apply it to some problems . let 's start off with something that we could have factored just to verify that it 's giving us the same answer . so let 's say we have x squared plus 4x minus 21 is equal to 0 . so in this situation -- let me do that in a different color -- a is equal to 1 , right ? the coefficient on the x squared term is 1. b is equal to 4 , the coefficient on the x-term . and then c is equal to negative 21 , the constant term . and let 's just plug it in the formula , so what do we get ? we get x , this tells us that x is going to be equal to negative b . negative b is negative 4 -- i put the negative sign in front of that -- negative b plus or minus the square root of b squared . b squared is 16 , right ? 4 squared is 16 , minus 4 times a , which is 1 , times c , which is negative 21 . so we can put a 21 out there and that negative sign will cancel out just like that with that -- since this is the first time we 're doing it , let me not skip too many steps . so negative 21 , just so you can see how it fit in , and then all of that over 2a . a is 1 , so all of that over 2 . so what does this simplify , or hopefully it simplifies ? so we get x is equal to negative 4 plus or minus the square root of -- let 's see we have a negative times a negative , that 's going to give us a positive . and we had 16 plus , let 's see this is 6 , 4 times 1 is 4 times 21 is 84 . 16 plus 84 is 100 . that 's nice . that 's a nice perfect square . all of that over 2 , and so this is going to be equal to negative 4 plus or minus 10 over 2 . we could just divide both of these terms by 2 right now . so this is equal to negative 4 divided by 2 is negative 2 plus or minus 10 divided by 2 is 5 . so that tells us that x could be equal to negative 2 plus 5 , which is 3 , or x could be equal to negative 2 minus 5 , which is negative 7 . so the quadratic formula seems to have given us an answer for this . you can verify just by substituting back in that these do work , or you could even just try to factor this right here . you say what two numbers when you take their product , you get negative 21 and when you take their sum you get positive 4 ? so you 'd get x plus 7 times x minus 3 is equal to negative 21 . notice 7 times negative 3 is negative 21 , 7 minus 3 is positive 4 . you would get x plus -- sorry it 's not negative -- 21 is equal to 0 . there should be a 0 there . so you get x plus 7 is equal to 0 , or x minus 3 is equal to 0 . x could be equal to negative 7 or x could be equal to 3 . so it definitely gives us the same answer as factoring , so you might say , hey why bother with this crazy mess ? and the reason we want to bother with this crazy mess is it 'll also work for problems that are hard to factor . and let 's do a couple of those , let 's do some hard-to-factor problems right now . so let 's scroll down to get some fresh real estate . let 's rewrite the formula again , just in case we have n't had it memorized yet . x is going to be equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . i 'll supply this to another problem . let 's say we have the equation 3x squared plus 6x is equal to negative 10 . well , the first thing we want to do is get it in the form where all of our terms or on the left-hand side , so let 's add 10 to both sides of this equation . we get 3x squared plus the 6x plus 10 is equal to 0 . and now we can use a quadratic formula . so let 's apply it here . so a is equal to 3 . that is a , this is b and this right here is c. so the quadratic formula tells us the solutions to this equation . the roots of this quadratic function , i guess we could call it . x is going to be equal to negative b. b is 6 , so negative 6 plus or minus the square root of b squared . b is 6 , so we get 6 squared minus 4 times a , which is 3 times c , which is 10 . let 's stretch out the radical little bit , all of that over 2 times a , 2 times 3 . so we get x is equal to negative 6 plus or minus the square root of 36 minus -- this is interesting -- minus 4 times 3 times 10 . so this is minus -- 4 times 3 times 10 . so this is minus 120 . all of that over 6 . so this is interesting , you might already realize why it 's interesting . what is this going to simplify to ? 36 minus 120 is what ? that 's 84 . we make this into a 10 , this will become an 11 , this is a 4 . it is 84 , so this is going to be equal to negative 6 plus or minus the square root of -- but not positive 84 , that 's if it 's 120 minus 36 . we have 36 minus 120 . it 's going to be negative 84 all of that 6 . so you might say , gee , this is crazy . what a this silly quadratic formula you 're introducing me to , sal ? it 's worthless . it just gives me a square root of a negative number . it 's not giving me an answer . and the reason why it 's not giving you an answer , at least an answer that you might want , is because this will have no real solutions . in the future , we 're going to introduce something called an imaginary number , which is a square root of a negative number , and then we can actually express this in terms of those numbers . so this actually does have solutions , but they involve imaginary numbers . so this actually has no real solutions , we 're taking the square root of a negative number . so the b squared with the b squared minus 4ac , if this term right here is negative , then you 're not going to have any real solutions . and let 's verify that for ourselves . let 's get our graphic calculator out and let 's graph this equation right here . so , let 's get the graphs that y is equal to -- that 's what i had there before -- 3x squared plus 6x plus 10 . so that 's the equation and we 're going to see where it intersects the x-axis . where does it equal 0 ? so let me graph it . notice , this thing just comes down and then goes back up . its vertex is sitting here above the x-axis and it 's upward-opening . it never intersects the x-axis . so at no point will this expression , will this function , equal 0 . at no point will y equal 0 on this graph . so once again , the quadratic formula seems to be working . let 's do one more example , you can never see enough examples here . and i want to do ones that are , you know , maybe not so obvious to factor . so let 's say we get negative 3x squared plus 12x plus 1 is equal to 0 . now let 's try to do it just having the quadratic formula in our brain . so the x 's that satisfy this equation are going to be negative b . this is b so negative b is negative 12 plus or minus the square root of b squared , of 144 , that 's b squared minus 4 times a , which is negative 3 times c , which is 1 , all of that over 2 times a , over 2 times negative 3 . so all of that over negative 6 , this is going to be equal to negative 12 plus or minus the square root of -- what is this ? it 's a negative times a negative so they cancel out . so i have 144 plus 12 , so that is 156 , right ? 144 plus 12 , all of that over negative 6 . now , i suspect we can simplify this 156 . we could maybe bring some things out of the radical sign . so let 's attempt to do that . so let 's do a prime factorization of 156 . sometimes , this is the hardest part , simplifying the radical . so 156 is the same thing as 2 times 78 . 78 is the same thing as 2 times what ? that 's 2 times 39 . so the square root of 156 is equal to the square root of 2 times 2 times 39 or we could say that 's the square root of 2 times 2 times the square root of 39 . and this , obviously , is just going to be the square root of 4 or this is the square root of 2 times 2 is just 2 . 2 square roots of 39 , if i did that properly , let 's see , 4 times 39 . yeah , it looks like it 's right . so this up here will simplify to negative 12 plus or minus 2 times the square root of 39 , all of that over negative 6 . now we can divide the numerator and the denominator maybe by 2 . so this will be equal to negative 6 plus or minus the square root of 39 over negative 3 . or we could separate these two terms out . we could say this is equal to negative 6 over negative 3 plus or minus the square root of 39 over negative 3 . now , this is just a 2 right here , right ? these cancel out , 6 divided by 3 is 2 , so we get 2 . and now notice , if this is plus and we use this minus sign , the plus will become negative and the negative will become positive . but it still does n't matter , right ? we could say minus or plus , that 's the same thing as plus or minus the square root of 39 nine over 3 . i think that 's about as simple as we can get this answered . i want to make a very clear point of what i did that last step . i did not forget about this negative sign . i just said it does n't matter . it 's going to turn the positive into the negative ; it 's going to turn the negative into the positive . let me rewrite this . so this right here can be rewritten as 2 plus the square root of 39 over negative 3 or 2 minus the square root of 39 over negative 3 , right ? that 's what the plus or minus means , it could be this or that or both of them , really . now in this situation , this negative 3 will turn into 2 minus the square root of 39 over 3 , right ? i 'm just taking this negative out . here the negative and the negative will become a positive , and you get 2 plus the square root of 39 over 3 , right ? a negative times a negative is a positive . so once again , you have 2 plus or minus the square of 39 over 3 . 2 plus or minus the square root of 39 over 3 are solutions to this equation right there . let verify . i 'm just curious what the graph looks like . so let 's just look at it . let me clear this . where is the clear button ? so we have negative 3 three squared plus 12x plus 1 and let 's graph it . let 's see where it intersects the x-axis . it goes up there and then back down again . so 2 plus or minus the square , you see -- the square root of 39 is going to be a little bit more than 6 , right ? because 36 is 6 squared . so it 's going be a little bit more than 6 , so this is going to be a little bit more than 2 . a little bit more than 6 divided by 2 is a little bit more than 2 . so you 're going to get one value that 's a little bit more than 4 and then another value that should be a little bit less than 1 . and that looks like the case , you have 1 , 2 , 3 , 4 . you have a value that 's pretty close to 4 , and then you have another value that is a little bit -- it looks close to 0 but maybe a little bit less than that . so anyway , hopefully you found this application of the quadratic formula helpful .
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we get 3x squared plus the 6x plus 10 is equal to 0 . and now we can use a quadratic formula . so let 's apply it here .
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how do you do quadratic equations with one solution with quadratic formula ?
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in this video , i 'm going to expose you to what is maybe one of at least the top five most useful formulas in mathematics . and if you 've seen many of my videos , you know that i 'm not a big fan of memorizing things . but i will recommend you memorize it with the caveat that you also remember how to prove it , because i do n't want you to just remember things and not know where they came from . but with that said , let me show you what i 'm talking about : it 's the quadratic formula . and as you might guess , it is to solve for the roots , or the zeroes of quadratic equations . so let 's speak in very general terms and i 'll show you some examples . so let 's say i have an equation of the form ax squared plus bx plus c is equal to 0 . you should recognize this . this is a quadratic equation where a , b and c are -- well , a is the coefficient on the x squared term or the second degree term , b is the coefficient on the x term and then c , is , you could imagine , the coefficient on the x to the zero term , or it 's the constant term . now , given that you have a general quadratic equation like this , the quadratic formula tells us that the solutions to this equation are x is equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . and i know it seems crazy and convoluted and hard for you to memorize right now , but as you get a lot more practice you 'll see that it actually is a pretty reasonable formula to stick in your brain someplace . and you might say , gee , this is a wacky formula , where did it come from ? and in the next video i 'm going to show you where it came from . but i want you to get used to using it first . but it really just came from completing the square on this equation right there . if you complete the square here , you 're actually going to get this solution and that is the quadratic formula , right there . so let 's apply it to some problems . let 's start off with something that we could have factored just to verify that it 's giving us the same answer . so let 's say we have x squared plus 4x minus 21 is equal to 0 . so in this situation -- let me do that in a different color -- a is equal to 1 , right ? the coefficient on the x squared term is 1. b is equal to 4 , the coefficient on the x-term . and then c is equal to negative 21 , the constant term . and let 's just plug it in the formula , so what do we get ? we get x , this tells us that x is going to be equal to negative b . negative b is negative 4 -- i put the negative sign in front of that -- negative b plus or minus the square root of b squared . b squared is 16 , right ? 4 squared is 16 , minus 4 times a , which is 1 , times c , which is negative 21 . so we can put a 21 out there and that negative sign will cancel out just like that with that -- since this is the first time we 're doing it , let me not skip too many steps . so negative 21 , just so you can see how it fit in , and then all of that over 2a . a is 1 , so all of that over 2 . so what does this simplify , or hopefully it simplifies ? so we get x is equal to negative 4 plus or minus the square root of -- let 's see we have a negative times a negative , that 's going to give us a positive . and we had 16 plus , let 's see this is 6 , 4 times 1 is 4 times 21 is 84 . 16 plus 84 is 100 . that 's nice . that 's a nice perfect square . all of that over 2 , and so this is going to be equal to negative 4 plus or minus 10 over 2 . we could just divide both of these terms by 2 right now . so this is equal to negative 4 divided by 2 is negative 2 plus or minus 10 divided by 2 is 5 . so that tells us that x could be equal to negative 2 plus 5 , which is 3 , or x could be equal to negative 2 minus 5 , which is negative 7 . so the quadratic formula seems to have given us an answer for this . you can verify just by substituting back in that these do work , or you could even just try to factor this right here . you say what two numbers when you take their product , you get negative 21 and when you take their sum you get positive 4 ? so you 'd get x plus 7 times x minus 3 is equal to negative 21 . notice 7 times negative 3 is negative 21 , 7 minus 3 is positive 4 . you would get x plus -- sorry it 's not negative -- 21 is equal to 0 . there should be a 0 there . so you get x plus 7 is equal to 0 , or x minus 3 is equal to 0 . x could be equal to negative 7 or x could be equal to 3 . so it definitely gives us the same answer as factoring , so you might say , hey why bother with this crazy mess ? and the reason we want to bother with this crazy mess is it 'll also work for problems that are hard to factor . and let 's do a couple of those , let 's do some hard-to-factor problems right now . so let 's scroll down to get some fresh real estate . let 's rewrite the formula again , just in case we have n't had it memorized yet . x is going to be equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . i 'll supply this to another problem . let 's say we have the equation 3x squared plus 6x is equal to negative 10 . well , the first thing we want to do is get it in the form where all of our terms or on the left-hand side , so let 's add 10 to both sides of this equation . we get 3x squared plus the 6x plus 10 is equal to 0 . and now we can use a quadratic formula . so let 's apply it here . so a is equal to 3 . that is a , this is b and this right here is c. so the quadratic formula tells us the solutions to this equation . the roots of this quadratic function , i guess we could call it . x is going to be equal to negative b. b is 6 , so negative 6 plus or minus the square root of b squared . b is 6 , so we get 6 squared minus 4 times a , which is 3 times c , which is 10 . let 's stretch out the radical little bit , all of that over 2 times a , 2 times 3 . so we get x is equal to negative 6 plus or minus the square root of 36 minus -- this is interesting -- minus 4 times 3 times 10 . so this is minus -- 4 times 3 times 10 . so this is minus 120 . all of that over 6 . so this is interesting , you might already realize why it 's interesting . what is this going to simplify to ? 36 minus 120 is what ? that 's 84 . we make this into a 10 , this will become an 11 , this is a 4 . it is 84 , so this is going to be equal to negative 6 plus or minus the square root of -- but not positive 84 , that 's if it 's 120 minus 36 . we have 36 minus 120 . it 's going to be negative 84 all of that 6 . so you might say , gee , this is crazy . what a this silly quadratic formula you 're introducing me to , sal ? it 's worthless . it just gives me a square root of a negative number . it 's not giving me an answer . and the reason why it 's not giving you an answer , at least an answer that you might want , is because this will have no real solutions . in the future , we 're going to introduce something called an imaginary number , which is a square root of a negative number , and then we can actually express this in terms of those numbers . so this actually does have solutions , but they involve imaginary numbers . so this actually has no real solutions , we 're taking the square root of a negative number . so the b squared with the b squared minus 4ac , if this term right here is negative , then you 're not going to have any real solutions . and let 's verify that for ourselves . let 's get our graphic calculator out and let 's graph this equation right here . so , let 's get the graphs that y is equal to -- that 's what i had there before -- 3x squared plus 6x plus 10 . so that 's the equation and we 're going to see where it intersects the x-axis . where does it equal 0 ? so let me graph it . notice , this thing just comes down and then goes back up . its vertex is sitting here above the x-axis and it 's upward-opening . it never intersects the x-axis . so at no point will this expression , will this function , equal 0 . at no point will y equal 0 on this graph . so once again , the quadratic formula seems to be working . let 's do one more example , you can never see enough examples here . and i want to do ones that are , you know , maybe not so obvious to factor . so let 's say we get negative 3x squared plus 12x plus 1 is equal to 0 . now let 's try to do it just having the quadratic formula in our brain . so the x 's that satisfy this equation are going to be negative b . this is b so negative b is negative 12 plus or minus the square root of b squared , of 144 , that 's b squared minus 4 times a , which is negative 3 times c , which is 1 , all of that over 2 times a , over 2 times negative 3 . so all of that over negative 6 , this is going to be equal to negative 12 plus or minus the square root of -- what is this ? it 's a negative times a negative so they cancel out . so i have 144 plus 12 , so that is 156 , right ? 144 plus 12 , all of that over negative 6 . now , i suspect we can simplify this 156 . we could maybe bring some things out of the radical sign . so let 's attempt to do that . so let 's do a prime factorization of 156 . sometimes , this is the hardest part , simplifying the radical . so 156 is the same thing as 2 times 78 . 78 is the same thing as 2 times what ? that 's 2 times 39 . so the square root of 156 is equal to the square root of 2 times 2 times 39 or we could say that 's the square root of 2 times 2 times the square root of 39 . and this , obviously , is just going to be the square root of 4 or this is the square root of 2 times 2 is just 2 . 2 square roots of 39 , if i did that properly , let 's see , 4 times 39 . yeah , it looks like it 's right . so this up here will simplify to negative 12 plus or minus 2 times the square root of 39 , all of that over negative 6 . now we can divide the numerator and the denominator maybe by 2 . so this will be equal to negative 6 plus or minus the square root of 39 over negative 3 . or we could separate these two terms out . we could say this is equal to negative 6 over negative 3 plus or minus the square root of 39 over negative 3 . now , this is just a 2 right here , right ? these cancel out , 6 divided by 3 is 2 , so we get 2 . and now notice , if this is plus and we use this minus sign , the plus will become negative and the negative will become positive . but it still does n't matter , right ? we could say minus or plus , that 's the same thing as plus or minus the square root of 39 nine over 3 . i think that 's about as simple as we can get this answered . i want to make a very clear point of what i did that last step . i did not forget about this negative sign . i just said it does n't matter . it 's going to turn the positive into the negative ; it 's going to turn the negative into the positive . let me rewrite this . so this right here can be rewritten as 2 plus the square root of 39 over negative 3 or 2 minus the square root of 39 over negative 3 , right ? that 's what the plus or minus means , it could be this or that or both of them , really . now in this situation , this negative 3 will turn into 2 minus the square root of 39 over 3 , right ? i 'm just taking this negative out . here the negative and the negative will become a positive , and you get 2 plus the square root of 39 over 3 , right ? a negative times a negative is a positive . so once again , you have 2 plus or minus the square of 39 over 3 . 2 plus or minus the square root of 39 over 3 are solutions to this equation right there . let verify . i 'm just curious what the graph looks like . so let 's just look at it . let me clear this . where is the clear button ? so we have negative 3 three squared plus 12x plus 1 and let 's graph it . let 's see where it intersects the x-axis . it goes up there and then back down again . so 2 plus or minus the square , you see -- the square root of 39 is going to be a little bit more than 6 , right ? because 36 is 6 squared . so it 's going be a little bit more than 6 , so this is going to be a little bit more than 2 . a little bit more than 6 divided by 2 is a little bit more than 2 . so you 're going to get one value that 's a little bit more than 4 and then another value that should be a little bit less than 1 . and that looks like the case , you have 1 , 2 , 3 , 4 . you have a value that 's pretty close to 4 , and then you have another value that is a little bit -- it looks close to 0 but maybe a little bit less than that . so anyway , hopefully you found this application of the quadratic formula helpful .
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that 's 2 times 39 . so the square root of 156 is equal to the square root of 2 times 2 times 39 or we could say that 's the square root of 2 times 2 times the square root of 39 . and this , obviously , is just going to be the square root of 4 or this is the square root of 2 times 2 is just 2 . 2 square roots of 39 , if i did that properly , let 's see , 4 times 39 .
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ca n't the b^2 inside the square root be simplified ?
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in this video , i 'm going to expose you to what is maybe one of at least the top five most useful formulas in mathematics . and if you 've seen many of my videos , you know that i 'm not a big fan of memorizing things . but i will recommend you memorize it with the caveat that you also remember how to prove it , because i do n't want you to just remember things and not know where they came from . but with that said , let me show you what i 'm talking about : it 's the quadratic formula . and as you might guess , it is to solve for the roots , or the zeroes of quadratic equations . so let 's speak in very general terms and i 'll show you some examples . so let 's say i have an equation of the form ax squared plus bx plus c is equal to 0 . you should recognize this . this is a quadratic equation where a , b and c are -- well , a is the coefficient on the x squared term or the second degree term , b is the coefficient on the x term and then c , is , you could imagine , the coefficient on the x to the zero term , or it 's the constant term . now , given that you have a general quadratic equation like this , the quadratic formula tells us that the solutions to this equation are x is equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . and i know it seems crazy and convoluted and hard for you to memorize right now , but as you get a lot more practice you 'll see that it actually is a pretty reasonable formula to stick in your brain someplace . and you might say , gee , this is a wacky formula , where did it come from ? and in the next video i 'm going to show you where it came from . but i want you to get used to using it first . but it really just came from completing the square on this equation right there . if you complete the square here , you 're actually going to get this solution and that is the quadratic formula , right there . so let 's apply it to some problems . let 's start off with something that we could have factored just to verify that it 's giving us the same answer . so let 's say we have x squared plus 4x minus 21 is equal to 0 . so in this situation -- let me do that in a different color -- a is equal to 1 , right ? the coefficient on the x squared term is 1. b is equal to 4 , the coefficient on the x-term . and then c is equal to negative 21 , the constant term . and let 's just plug it in the formula , so what do we get ? we get x , this tells us that x is going to be equal to negative b . negative b is negative 4 -- i put the negative sign in front of that -- negative b plus or minus the square root of b squared . b squared is 16 , right ? 4 squared is 16 , minus 4 times a , which is 1 , times c , which is negative 21 . so we can put a 21 out there and that negative sign will cancel out just like that with that -- since this is the first time we 're doing it , let me not skip too many steps . so negative 21 , just so you can see how it fit in , and then all of that over 2a . a is 1 , so all of that over 2 . so what does this simplify , or hopefully it simplifies ? so we get x is equal to negative 4 plus or minus the square root of -- let 's see we have a negative times a negative , that 's going to give us a positive . and we had 16 plus , let 's see this is 6 , 4 times 1 is 4 times 21 is 84 . 16 plus 84 is 100 . that 's nice . that 's a nice perfect square . all of that over 2 , and so this is going to be equal to negative 4 plus or minus 10 over 2 . we could just divide both of these terms by 2 right now . so this is equal to negative 4 divided by 2 is negative 2 plus or minus 10 divided by 2 is 5 . so that tells us that x could be equal to negative 2 plus 5 , which is 3 , or x could be equal to negative 2 minus 5 , which is negative 7 . so the quadratic formula seems to have given us an answer for this . you can verify just by substituting back in that these do work , or you could even just try to factor this right here . you say what two numbers when you take their product , you get negative 21 and when you take their sum you get positive 4 ? so you 'd get x plus 7 times x minus 3 is equal to negative 21 . notice 7 times negative 3 is negative 21 , 7 minus 3 is positive 4 . you would get x plus -- sorry it 's not negative -- 21 is equal to 0 . there should be a 0 there . so you get x plus 7 is equal to 0 , or x minus 3 is equal to 0 . x could be equal to negative 7 or x could be equal to 3 . so it definitely gives us the same answer as factoring , so you might say , hey why bother with this crazy mess ? and the reason we want to bother with this crazy mess is it 'll also work for problems that are hard to factor . and let 's do a couple of those , let 's do some hard-to-factor problems right now . so let 's scroll down to get some fresh real estate . let 's rewrite the formula again , just in case we have n't had it memorized yet . x is going to be equal to negative b plus or minus the square root of b squared minus 4ac , all of that over 2a . i 'll supply this to another problem . let 's say we have the equation 3x squared plus 6x is equal to negative 10 . well , the first thing we want to do is get it in the form where all of our terms or on the left-hand side , so let 's add 10 to both sides of this equation . we get 3x squared plus the 6x plus 10 is equal to 0 . and now we can use a quadratic formula . so let 's apply it here . so a is equal to 3 . that is a , this is b and this right here is c. so the quadratic formula tells us the solutions to this equation . the roots of this quadratic function , i guess we could call it . x is going to be equal to negative b. b is 6 , so negative 6 plus or minus the square root of b squared . b is 6 , so we get 6 squared minus 4 times a , which is 3 times c , which is 10 . let 's stretch out the radical little bit , all of that over 2 times a , 2 times 3 . so we get x is equal to negative 6 plus or minus the square root of 36 minus -- this is interesting -- minus 4 times 3 times 10 . so this is minus -- 4 times 3 times 10 . so this is minus 120 . all of that over 6 . so this is interesting , you might already realize why it 's interesting . what is this going to simplify to ? 36 minus 120 is what ? that 's 84 . we make this into a 10 , this will become an 11 , this is a 4 . it is 84 , so this is going to be equal to negative 6 plus or minus the square root of -- but not positive 84 , that 's if it 's 120 minus 36 . we have 36 minus 120 . it 's going to be negative 84 all of that 6 . so you might say , gee , this is crazy . what a this silly quadratic formula you 're introducing me to , sal ? it 's worthless . it just gives me a square root of a negative number . it 's not giving me an answer . and the reason why it 's not giving you an answer , at least an answer that you might want , is because this will have no real solutions . in the future , we 're going to introduce something called an imaginary number , which is a square root of a negative number , and then we can actually express this in terms of those numbers . so this actually does have solutions , but they involve imaginary numbers . so this actually has no real solutions , we 're taking the square root of a negative number . so the b squared with the b squared minus 4ac , if this term right here is negative , then you 're not going to have any real solutions . and let 's verify that for ourselves . let 's get our graphic calculator out and let 's graph this equation right here . so , let 's get the graphs that y is equal to -- that 's what i had there before -- 3x squared plus 6x plus 10 . so that 's the equation and we 're going to see where it intersects the x-axis . where does it equal 0 ? so let me graph it . notice , this thing just comes down and then goes back up . its vertex is sitting here above the x-axis and it 's upward-opening . it never intersects the x-axis . so at no point will this expression , will this function , equal 0 . at no point will y equal 0 on this graph . so once again , the quadratic formula seems to be working . let 's do one more example , you can never see enough examples here . and i want to do ones that are , you know , maybe not so obvious to factor . so let 's say we get negative 3x squared plus 12x plus 1 is equal to 0 . now let 's try to do it just having the quadratic formula in our brain . so the x 's that satisfy this equation are going to be negative b . this is b so negative b is negative 12 plus or minus the square root of b squared , of 144 , that 's b squared minus 4 times a , which is negative 3 times c , which is 1 , all of that over 2 times a , over 2 times negative 3 . so all of that over negative 6 , this is going to be equal to negative 12 plus or minus the square root of -- what is this ? it 's a negative times a negative so they cancel out . so i have 144 plus 12 , so that is 156 , right ? 144 plus 12 , all of that over negative 6 . now , i suspect we can simplify this 156 . we could maybe bring some things out of the radical sign . so let 's attempt to do that . so let 's do a prime factorization of 156 . sometimes , this is the hardest part , simplifying the radical . so 156 is the same thing as 2 times 78 . 78 is the same thing as 2 times what ? that 's 2 times 39 . so the square root of 156 is equal to the square root of 2 times 2 times 39 or we could say that 's the square root of 2 times 2 times the square root of 39 . and this , obviously , is just going to be the square root of 4 or this is the square root of 2 times 2 is just 2 . 2 square roots of 39 , if i did that properly , let 's see , 4 times 39 . yeah , it looks like it 's right . so this up here will simplify to negative 12 plus or minus 2 times the square root of 39 , all of that over negative 6 . now we can divide the numerator and the denominator maybe by 2 . so this will be equal to negative 6 plus or minus the square root of 39 over negative 3 . or we could separate these two terms out . we could say this is equal to negative 6 over negative 3 plus or minus the square root of 39 over negative 3 . now , this is just a 2 right here , right ? these cancel out , 6 divided by 3 is 2 , so we get 2 . and now notice , if this is plus and we use this minus sign , the plus will become negative and the negative will become positive . but it still does n't matter , right ? we could say minus or plus , that 's the same thing as plus or minus the square root of 39 nine over 3 . i think that 's about as simple as we can get this answered . i want to make a very clear point of what i did that last step . i did not forget about this negative sign . i just said it does n't matter . it 's going to turn the positive into the negative ; it 's going to turn the negative into the positive . let me rewrite this . so this right here can be rewritten as 2 plus the square root of 39 over negative 3 or 2 minus the square root of 39 over negative 3 , right ? that 's what the plus or minus means , it could be this or that or both of them , really . now in this situation , this negative 3 will turn into 2 minus the square root of 39 over 3 , right ? i 'm just taking this negative out . here the negative and the negative will become a positive , and you get 2 plus the square root of 39 over 3 , right ? a negative times a negative is a positive . so once again , you have 2 plus or minus the square of 39 over 3 . 2 plus or minus the square root of 39 over 3 are solutions to this equation right there . let verify . i 'm just curious what the graph looks like . so let 's just look at it . let me clear this . where is the clear button ? so we have negative 3 three squared plus 12x plus 1 and let 's graph it . let 's see where it intersects the x-axis . it goes up there and then back down again . so 2 plus or minus the square , you see -- the square root of 39 is going to be a little bit more than 6 , right ? because 36 is 6 squared . so it 's going be a little bit more than 6 , so this is going to be a little bit more than 2 . a little bit more than 6 divided by 2 is a little bit more than 2 . so you 're going to get one value that 's a little bit more than 4 and then another value that should be a little bit less than 1 . and that looks like the case , you have 1 , 2 , 3 , 4 . you have a value that 's pretty close to 4 , and then you have another value that is a little bit -- it looks close to 0 but maybe a little bit less than that . so anyway , hopefully you found this application of the quadratic formula helpful .
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and now notice , if this is plus and we use this minus sign , the plus will become negative and the negative will become positive . but it still does n't matter , right ? we could say minus or plus , that 's the same thing as plus or minus the square root of 39 nine over 3 .
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4 why do you still have two answers does n't the + or - sign disappear after the sqr root is found ?
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