video_title stringlengths 25 104 | Sentence stringlengths 91 1.69k |
|---|---|
The Pythagorean theorem intro Right triangles and trigonometry Geometry Khan Academy.mp3 | So 108 is the same thing as 2 times 54, which is the same thing as 2 times 27, which is the same thing as 3 times 9. So we have the square root of 108 is the same thing as the square root of 2 times 2 times, actually I'm not done, 9 can be factorized into 3 times 3. So it's 2 times 2 times 3 times 3 times 3. And so we have a couple of perfect squares in here. Let me rewrite it a little bit neater. And this is all an exercise in simplifying radicals that you will bump into a lot while doing the Pythagorean theorem, so it doesn't hurt to do it right here. So this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there. |
The Pythagorean theorem intro Right triangles and trigonometry Geometry Khan Academy.mp3 | And so we have a couple of perfect squares in here. Let me rewrite it a little bit neater. And this is all an exercise in simplifying radicals that you will bump into a lot while doing the Pythagorean theorem, so it doesn't hurt to do it right here. So this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there. And this is the same thing, and you know you wouldn't have to do all of this in your head, well all of this on paper, you could do it in your head. What is this? 2 times 2 is 4, 4 times 9, this is 36. |
The Pythagorean theorem intro Right triangles and trigonometry Geometry Khan Academy.mp3 | So this is the same thing as the square root of 2 times 2 times 3 times 3 times the square root of that last 3 right over there. And this is the same thing, and you know you wouldn't have to do all of this in your head, well all of this on paper, you could do it in your head. What is this? 2 times 2 is 4, 4 times 9, this is 36. This is the square root of 36 times the square root of 3. The principal root of 36 is 6. So this simplifies to 6 square roots of 3. |
Special products of the form (x+a)(x-a) Algebra I High School Math Khan Academy.mp3 | Let's see if we can figure out what x plus three times x minus three is, and I encourage you to pause the video and see if you can work this out. Well, one way to tackle it is the way that we've always tackled when we multiply binomials is just apply the distributive property twice. So first, we could take this entire yellow x plus three and multiply it times each of these two terms. So first, we can multiply it times this x, so that's going to be x times x plus three, and then we are going to multiply it times, we could say this negative three. So we could write minus three times, now that's going to be multiplied by x plus three again, x plus three, and then we apply the distributive property one more time, where we take this magenta x and we distribute it across this x plus three, so x times x is x squared, x times three is three x, and then we do it on this side. Negative three times x is negative three x, and negative three times three is negative nine. And what does this simplify to? |
Special products of the form (x+a)(x-a) Algebra I High School Math Khan Academy.mp3 | So first, we can multiply it times this x, so that's going to be x times x plus three, and then we are going to multiply it times, we could say this negative three. So we could write minus three times, now that's going to be multiplied by x plus three again, x plus three, and then we apply the distributive property one more time, where we take this magenta x and we distribute it across this x plus three, so x times x is x squared, x times three is three x, and then we do it on this side. Negative three times x is negative three x, and negative three times three is negative nine. And what does this simplify to? Well, we're gonna get x squared, and we have three x minus three x, so these two characters cancel out, and we are just left with x squared minus nine. And you might see a little pattern here. Notice, I added three and then I subtracted three, and I got this, I got the x squared, and then if you take three and multiply it by negative three, you are going to get a negative nine. |
Special products of the form (x+a)(x-a) Algebra I High School Math Khan Academy.mp3 | And what does this simplify to? Well, we're gonna get x squared, and we have three x minus three x, so these two characters cancel out, and we are just left with x squared minus nine. And you might see a little pattern here. Notice, I added three and then I subtracted three, and I got this, I got the x squared, and then if you take three and multiply it by negative three, you are going to get a negative nine. And notice, the middle terms canceled out, and one thing you might ask is, well, will that always be the case if we add a number and then we subtract that same number like that? And we could try it out. Let's talk in general terms. |
Special products of the form (x+a)(x-a) Algebra I High School Math Khan Academy.mp3 | Notice, I added three and then I subtracted three, and I got this, I got the x squared, and then if you take three and multiply it by negative three, you are going to get a negative nine. And notice, the middle terms canceled out, and one thing you might ask is, well, will that always be the case if we add a number and then we subtract that same number like that? And we could try it out. Let's talk in general terms. So if we, instead of doing x plus three times x minus three, we could write the same thing as, instead of three, let's just say you have x plus, x plus a times x minus a, times x minus a. And I encourage you to pause this video and work it all out. Just assume a is some number, like three or some other number, and apply the distributive property twice and see what you get. |
Special products of the form (x+a)(x-a) Algebra I High School Math Khan Academy.mp3 | Let's talk in general terms. So if we, instead of doing x plus three times x minus three, we could write the same thing as, instead of three, let's just say you have x plus, x plus a times x minus a, times x minus a. And I encourage you to pause this video and work it all out. Just assume a is some number, like three or some other number, and apply the distributive property twice and see what you get. Well, let's work through it. So first we can distribute this yellow x plus a onto the x and the negative a. So x plus a times x, or we could say x times x plus a. |
Special products of the form (x+a)(x-a) Algebra I High School Math Khan Academy.mp3 | Just assume a is some number, like three or some other number, and apply the distributive property twice and see what you get. Well, let's work through it. So first we can distribute this yellow x plus a onto the x and the negative a. So x plus a times x, or we could say x times x plus a. So that's going to be, that's going to be x times x plus a, and then we're going to have minus a, or this negative a times x plus a. So minus, and then we're going to have this minus a times x plus a, times x plus a, times x plus a. Notice, all I did is I distributed this yellow, I just distributed this big chunk of this expression, I just distributed it onto the x and onto this negative a. I'm multiplying it times the x and I'm multiplying it by the negative a. |
Special products of the form (x+a)(x-a) Algebra I High School Math Khan Academy.mp3 | So x plus a times x, or we could say x times x plus a. So that's going to be, that's going to be x times x plus a, and then we're going to have minus a, or this negative a times x plus a. So minus, and then we're going to have this minus a times x plus a, times x plus a, times x plus a. Notice, all I did is I distributed this yellow, I just distributed this big chunk of this expression, I just distributed it onto the x and onto this negative a. I'm multiplying it times the x and I'm multiplying it by the negative a. And now we can apply the distributive property again. X times x is x squared. X times a is ax. |
Special products of the form (x+a)(x-a) Algebra I High School Math Khan Academy.mp3 | Notice, all I did is I distributed this yellow, I just distributed this big chunk of this expression, I just distributed it onto the x and onto this negative a. I'm multiplying it times the x and I'm multiplying it by the negative a. And now we can apply the distributive property again. X times x is x squared. X times a is ax. And then we get negative a times x is negative ax. And then negative a times a is negative a squared. And notice, regardless of my choice of a, I'm going to have ax and then minus ax. |
Special products of the form (x+a)(x-a) Algebra I High School Math Khan Academy.mp3 | X times a is ax. And then we get negative a times x is negative ax. And then negative a times a is negative a squared. And notice, regardless of my choice of a, I'm going to have ax and then minus ax. So this is always going to cancel out. It didn't just work for the case when a was three. For any a, if I have a times x and then I subtract a times x, that's just going to cancel out. |
Special products of the form (x+a)(x-a) Algebra I High School Math Khan Academy.mp3 | And notice, regardless of my choice of a, I'm going to have ax and then minus ax. So this is always going to cancel out. It didn't just work for the case when a was three. For any a, if I have a times x and then I subtract a times x, that's just going to cancel out. So this is just going to cancel out. And what are we going to be left with? We are going to be left with x squared minus a squared. |
Special products of the form (x+a)(x-a) Algebra I High School Math Khan Academy.mp3 | For any a, if I have a times x and then I subtract a times x, that's just going to cancel out. So this is just going to cancel out. And what are we going to be left with? We are going to be left with x squared minus a squared. X squared minus a squared. And you could view this as a special case. When you have something x plus something times x minus that same something, it's going to be x squared minus that something squared. |
Special products of the form (x+a)(x-a) Algebra I High School Math Khan Academy.mp3 | We are going to be left with x squared minus a squared. X squared minus a squared. And you could view this as a special case. When you have something x plus something times x minus that same something, it's going to be x squared minus that something squared. And this is a good one to know in general. This is a good one to know in general. And we could use it to quickly figure out the products of other binomials that fit this pattern here. |
Special products of the form (x+a)(x-a) Algebra I High School Math Khan Academy.mp3 | When you have something x plus something times x minus that same something, it's going to be x squared minus that something squared. And this is a good one to know in general. This is a good one to know in general. And we could use it to quickly figure out the products of other binomials that fit this pattern here. So if I were to say, quick, what is x plus 10 times x minus 10? Well, you could say, all right, this fits the pattern. It's x plus a times x minus a. |
Special products of the form (x+a)(x-a) Algebra I High School Math Khan Academy.mp3 | And we could use it to quickly figure out the products of other binomials that fit this pattern here. So if I were to say, quick, what is x plus 10 times x minus 10? Well, you could say, all right, this fits the pattern. It's x plus a times x minus a. So it's going to be x squared minus a squared. If a is 10, a squared is going to be 100. So you can do it really quick once you recognize the pattern. |
Percent word problem example 2 Decimals Pre-Algebra Khan Academy.mp3 | So 13 out of every 20 are recycled. So 13 20, it's 13 over 20, could also be viewed as 13, 13 divided by 20, or 13 divided by 20. And if we do this, we'll get a decimal, it's fairly straightforward to convert that decimal into a percentage. So 13 divided by 20, we have the smaller number, in this case, being divided by the larger number. So we're going to get a value less than one. Since we're gonna get a value less than one, let's put a decimal right over here, and let's add a couple of zeros, as many zeros as we would need. And we could say, hey look, 20 goes into 13, zero times, zero times 20 is zero, and then 13 minus zero is 13, now you bring down a zero. |
Percent word problem example 2 Decimals Pre-Algebra Khan Academy.mp3 | So 13 divided by 20, we have the smaller number, in this case, being divided by the larger number. So we're going to get a value less than one. Since we're gonna get a value less than one, let's put a decimal right over here, and let's add a couple of zeros, as many zeros as we would need. And we could say, hey look, 20 goes into 13, zero times, zero times 20 is zero, and then 13 minus zero is 13, now you bring down a zero. 20 goes into 130, let's see, it goes in, five times 20 is 100, so six times 20 is 120, so it's gonna go six times. Six times 20 is 120, you subtract, you get a 10, let's bring down another zero. 20 goes into 100 five times, five times 20 is 100, and we are done. |
Percent word problem example 2 Decimals Pre-Algebra Khan Academy.mp3 | And we could say, hey look, 20 goes into 13, zero times, zero times 20 is zero, and then 13 minus zero is 13, now you bring down a zero. 20 goes into 130, let's see, it goes in, five times 20 is 100, so six times 20 is 120, so it's gonna go six times. Six times 20 is 120, you subtract, you get a 10, let's bring down another zero. 20 goes into 100 five times, five times 20 is 100, and we are done. So this written as a decimal, this written as a decimal is 0.65. So as a decimal, it's 0.65, and if you wanna write it as a percentage, you essentially multiply this by 100, or another way you could say is you shift the decimal over two spots to the right. So this is going to be equal to 65%, 65%. |
Percent word problem example 2 Decimals Pre-Algebra Khan Academy.mp3 | 20 goes into 100 five times, five times 20 is 100, and we are done. So this written as a decimal, this written as a decimal is 0.65. So as a decimal, it's 0.65, and if you wanna write it as a percentage, you essentially multiply this by 100, or another way you could say is you shift the decimal over two spots to the right. So this is going to be equal to 65%, 65%. Now, there's another way you could have done it, is you could have said, look, percent literally means per 100, so 13 out of 20, 13 out of 20 is going to be equal to what, over, is going to be equal to what over 100? Well, to go from 20 to 100, you forget the denominator, to go from 20 to 100, you would multiply by five, so let's multiply the numerator by five as well. And 13 times five, let's see, that's 15 plus 50, which is 65. |
Similarity postulates Similarity Geometry Khan Academy.mp3 | Let's say we have triangle ABC. It looks something like this. ABC. I want to think about the minimum amount of information. I want to come up with a couple of postulates that we can use to determine whether another triangle is similar to triangle ABC. So we already know that if all three angles, all three of the corresponding angles are congruent to the corresponding angles on ABC, then we know that we're dealing with congruent triangles. For example, if this is 30 degrees, this angle is 90 degrees, and this angle right over here is 60 degrees, and we have another triangle that looks like this, that looks like this. |
Similarity postulates Similarity Geometry Khan Academy.mp3 | I want to think about the minimum amount of information. I want to come up with a couple of postulates that we can use to determine whether another triangle is similar to triangle ABC. So we already know that if all three angles, all three of the corresponding angles are congruent to the corresponding angles on ABC, then we know that we're dealing with congruent triangles. For example, if this is 30 degrees, this angle is 90 degrees, and this angle right over here is 60 degrees, and we have another triangle that looks like this, that looks like this. It's clearly a smaller triangle, but its corresponding angles, so this is 30 degrees, this is 90 degrees, and this is 60 degrees, we know that XYZ, in this case, is going to be similar to ABC. We would know from this, because corresponding angles are congruent, we would know that triangle ABC is similar to triangle XYZ. You've got to get the order right to make sure that you have the right corresponding angles. |
Similarity postulates Similarity Geometry Khan Academy.mp3 | For example, if this is 30 degrees, this angle is 90 degrees, and this angle right over here is 60 degrees, and we have another triangle that looks like this, that looks like this. It's clearly a smaller triangle, but its corresponding angles, so this is 30 degrees, this is 90 degrees, and this is 60 degrees, we know that XYZ, in this case, is going to be similar to ABC. We would know from this, because corresponding angles are congruent, we would know that triangle ABC is similar to triangle XYZ. You've got to get the order right to make sure that you have the right corresponding angles. Y corresponds to the 90 degree angle, X corresponds to the 30 degree angle, A corresponds to the 30 degree angle, so A and X are the first two things, B and Y, which are the 90 degrees, are the second two, and then Z is the last one. That's what we know already, if you have three angles. But do you need three angles? |
Similarity postulates Similarity Geometry Khan Academy.mp3 | You've got to get the order right to make sure that you have the right corresponding angles. Y corresponds to the 90 degree angle, X corresponds to the 30 degree angle, A corresponds to the 30 degree angle, so A and X are the first two things, B and Y, which are the 90 degrees, are the second two, and then Z is the last one. That's what we know already, if you have three angles. But do you need three angles? If we only knew two of the angles, would that be enough? Sure, because if you know two angles for a triangle, you know the third. For example, if I have another triangle, if I have a triangle that looks like this, and if I told you that only two of the corresponding angles are congruent, maybe this angle right here is congruent to this angle, and that angle right there is congruent to that angle. |
Similarity postulates Similarity Geometry Khan Academy.mp3 | But do you need three angles? If we only knew two of the angles, would that be enough? Sure, because if you know two angles for a triangle, you know the third. For example, if I have another triangle, if I have a triangle that looks like this, and if I told you that only two of the corresponding angles are congruent, maybe this angle right here is congruent to this angle, and that angle right there is congruent to that angle. Is that enough to say that these two triangles are similar? Sure, because in a triangle, if you know two of the angles, then you know what the last angle has to be. If you know that this is 30 and you know that that is 90, then you know that this angle has to be 60 degrees. |
Similarity postulates Similarity Geometry Khan Academy.mp3 | For example, if I have another triangle, if I have a triangle that looks like this, and if I told you that only two of the corresponding angles are congruent, maybe this angle right here is congruent to this angle, and that angle right there is congruent to that angle. Is that enough to say that these two triangles are similar? Sure, because in a triangle, if you know two of the angles, then you know what the last angle has to be. If you know that this is 30 and you know that that is 90, then you know that this angle has to be 60 degrees. Whatever these two angles are, subtract them from 180, and that's going to be this angle. In general, in order to show similarity, you don't have to show three corresponding angles are congruent. You really just have to show two. |
Similarity postulates Similarity Geometry Khan Academy.mp3 | If you know that this is 30 and you know that that is 90, then you know that this angle has to be 60 degrees. Whatever these two angles are, subtract them from 180, and that's going to be this angle. In general, in order to show similarity, you don't have to show three corresponding angles are congruent. You really just have to show two. This will be the first of our similarity postulates. We've called it angle-angle. If you could show that two corresponding angles are congruent, then we're dealing with similar triangles. |
Similarity postulates Similarity Geometry Khan Academy.mp3 | You really just have to show two. This will be the first of our similarity postulates. We've called it angle-angle. If you could show that two corresponding angles are congruent, then we're dealing with similar triangles. For example, just to put some numbers here, if this was 30 degrees, and we know that on this triangle this is 90 degrees right over here, we know that this triangle right over here is similar to that one there. You can really just go to the third angle in a pretty straightforward way. You say, hey, this third angle is 60 degrees, so all three angles are the same. |
Similarity postulates Similarity Geometry Khan Academy.mp3 | If you could show that two corresponding angles are congruent, then we're dealing with similar triangles. For example, just to put some numbers here, if this was 30 degrees, and we know that on this triangle this is 90 degrees right over here, we know that this triangle right over here is similar to that one there. You can really just go to the third angle in a pretty straightforward way. You say, hey, this third angle is 60 degrees, so all three angles are the same. That's one of our constraints for similarity. The other thing we know about similarity is that the ratio between all of the sides are going to be the same. For example, if we have another triangle right over here, let me draw another triangle. |
Similarity postulates Similarity Geometry Khan Academy.mp3 | You say, hey, this third angle is 60 degrees, so all three angles are the same. That's one of our constraints for similarity. The other thing we know about similarity is that the ratio between all of the sides are going to be the same. For example, if we have another triangle right over here, let me draw another triangle. I'll call this triangle x, y, and z. Let's say that we know that the ratio between AB and xy, we know that AB over xy, so the ratio between this side and this side, notice we're not saying that they're congruent, we're just saying that they're ratio. We're looking at the ratio now. |
Similarity postulates Similarity Geometry Khan Academy.mp3 | For example, if we have another triangle right over here, let me draw another triangle. I'll call this triangle x, y, and z. Let's say that we know that the ratio between AB and xy, we know that AB over xy, so the ratio between this side and this side, notice we're not saying that they're congruent, we're just saying that they're ratio. We're looking at the ratio now. We're saying AB over xy, let's say that that is equal to BC over yz, and that is equal to AC over xz. Once again, this is one of the ways that we say, hey, this means similarity. If we have all three corresponding sides, the ratio between all three corresponding sides are the same, then we know we are dealing with similar triangles. |
Similarity postulates Similarity Geometry Khan Academy.mp3 | We're looking at the ratio now. We're saying AB over xy, let's say that that is equal to BC over yz, and that is equal to AC over xz. Once again, this is one of the ways that we say, hey, this means similarity. If we have all three corresponding sides, the ratio between all three corresponding sides are the same, then we know we are dealing with similar triangles. This is what we call side-side-side similarity. You don't want to get these confused with side-side-side congruence. These are all of our similarity postulates, or axioms, or things that we're going to assume and then we're going to build off of them to solve problems and prove other things. |
Similarity postulates Similarity Geometry Khan Academy.mp3 | If we have all three corresponding sides, the ratio between all three corresponding sides are the same, then we know we are dealing with similar triangles. This is what we call side-side-side similarity. You don't want to get these confused with side-side-side congruence. These are all of our similarity postulates, or axioms, or things that we're going to assume and then we're going to build off of them to solve problems and prove other things. Side-side-side, when we're talking about congruence, means that the corresponding sides are congruent. Side-side-side for similarity, we're saying that the ratio between corresponding sides are going to be the same. For example, if this right over here is 10, let's say this is 60, this right over here is 30, and this right over here is 30 square roots of 3. |
Similarity postulates Similarity Geometry Khan Academy.mp3 | These are all of our similarity postulates, or axioms, or things that we're going to assume and then we're going to build off of them to solve problems and prove other things. Side-side-side, when we're talking about congruence, means that the corresponding sides are congruent. Side-side-side for similarity, we're saying that the ratio between corresponding sides are going to be the same. For example, if this right over here is 10, let's say this is 60, this right over here is 30, and this right over here is 30 square roots of 3. I just made those numbers right because we will soon learn what typical ratios are of the sides of 30, 60, 90 triangles. Let's say this one over here is 6, 3, and 3 square roots of 3. Notice, AB over xy, 30 square roots of 3 over 3 square roots of 3, this will be 10. |
Similarity postulates Similarity Geometry Khan Academy.mp3 | For example, if this right over here is 10, let's say this is 60, this right over here is 30, and this right over here is 30 square roots of 3. I just made those numbers right because we will soon learn what typical ratios are of the sides of 30, 60, 90 triangles. Let's say this one over here is 6, 3, and 3 square roots of 3. Notice, AB over xy, 30 square roots of 3 over 3 square roots of 3, this will be 10. What is BC over xy? 30 divided by 3 is 10. What is 60 divided by 6? |
Similarity postulates Similarity Geometry Khan Academy.mp3 | Notice, AB over xy, 30 square roots of 3 over 3 square roots of 3, this will be 10. What is BC over xy? 30 divided by 3 is 10. What is 60 divided by 6? AC over xz, that's going to be 10. In general, to go from the corresponding side here to the corresponding side there, we always multiply by 10 on every side. We're not saying the sides are the same for this side-side-side for similarity. |
Similarity postulates Similarity Geometry Khan Academy.mp3 | What is 60 divided by 6? AC over xz, that's going to be 10. In general, to go from the corresponding side here to the corresponding side there, we always multiply by 10 on every side. We're not saying the sides are the same for this side-side-side for similarity. We're saying that we're really just scaling them up by the same amount. Or, another way to think about it, the ratio between corresponding sides are the same. Now what about if we had, let's start another triangle right over here. |
Similarity postulates Similarity Geometry Khan Academy.mp3 | We're not saying the sides are the same for this side-side-side for similarity. We're saying that we're really just scaling them up by the same amount. Or, another way to think about it, the ratio between corresponding sides are the same. Now what about if we had, let's start another triangle right over here. Let me draw it like this. Actually, I want to leave this here so we can have our list. Let me draw another triangle ABC. |
Similarity postulates Similarity Geometry Khan Academy.mp3 | Now what about if we had, let's start another triangle right over here. Let me draw it like this. Actually, I want to leave this here so we can have our list. Let me draw another triangle ABC. Let's draw another triangle ABC. This is A, B, and C. Let's say that we know that this side, when we go to another triangle, we know that xy is AB multiplied by some constant. I can write it over here. |
Similarity postulates Similarity Geometry Khan Academy.mp3 | Let me draw another triangle ABC. Let's draw another triangle ABC. This is A, B, and C. Let's say that we know that this side, when we go to another triangle, we know that xy is AB multiplied by some constant. I can write it over here. xy is equal to some constant times AB. Actually, let me make xy bigger. It doesn't have to be. |
Similarity postulates Similarity Geometry Khan Academy.mp3 | I can write it over here. xy is equal to some constant times AB. Actually, let me make xy bigger. It doesn't have to be. That constant could be less than 1, in which case it would be a smaller value. Let me just do it that way. Let me just make xy look a little bit bigger. |
Similarity postulates Similarity Geometry Khan Academy.mp3 | It doesn't have to be. That constant could be less than 1, in which case it would be a smaller value. Let me just do it that way. Let me just make xy look a little bit bigger. Let's say that this is x and that is y. Let's say that we know that xy over AB is equal to some constant. Or, if you multiply both sides by AB, you would get xy is some scaled-up version of AB. |
Similarity postulates Similarity Geometry Khan Academy.mp3 | Let me just make xy look a little bit bigger. Let's say that this is x and that is y. Let's say that we know that xy over AB is equal to some constant. Or, if you multiply both sides by AB, you would get xy is some scaled-up version of AB. Maybe AB is 5, xy is 10, then our constant would be 2. We scaled it up by a factor of 2. Let's say we also know that angle ABC is congruent to angle XYZ. |
Similarity postulates Similarity Geometry Khan Academy.mp3 | Or, if you multiply both sides by AB, you would get xy is some scaled-up version of AB. Maybe AB is 5, xy is 10, then our constant would be 2. We scaled it up by a factor of 2. Let's say we also know that angle ABC is congruent to angle XYZ. I'll add another point over here. Let me draw another side right over here. This is Z. |
Similarity postulates Similarity Geometry Khan Academy.mp3 | Let's say we also know that angle ABC is congruent to angle XYZ. I'll add another point over here. Let me draw another side right over here. This is Z. Let's say we also know that angle ABC is congruent to XYZ. Let's say we know that the ratio between BC and YZ is also this constant. The ratio between BC and YZ is also equal to the same constant. |
Similarity postulates Similarity Geometry Khan Academy.mp3 | This is Z. Let's say we also know that angle ABC is congruent to XYZ. Let's say we know that the ratio between BC and YZ is also this constant. The ratio between BC and YZ is also equal to the same constant. In the example where this is 5 and 10, maybe this is 3 and 6. The constant, we're doubling the length of the side. Is this triangle, is triangle XYZ going to be similar? |
Similarity postulates Similarity Geometry Khan Academy.mp3 | The ratio between BC and YZ is also equal to the same constant. In the example where this is 5 and 10, maybe this is 3 and 6. The constant, we're doubling the length of the side. Is this triangle, is triangle XYZ going to be similar? If you think about it, if you say that this is some multiple, if XY is the same multiple of AB as YZ is the multiple of BC, and the angle in between is congruent, there's only one triangle we can set up over here. We're only constrained to one triangle right over here. We're completely constraining the length of this side. |
Similarity postulates Similarity Geometry Khan Academy.mp3 | Is this triangle, is triangle XYZ going to be similar? If you think about it, if you say that this is some multiple, if XY is the same multiple of AB as YZ is the multiple of BC, and the angle in between is congruent, there's only one triangle we can set up over here. We're only constrained to one triangle right over here. We're completely constraining the length of this side. The length of this side is going to have to be that same scale as that over there. We call that side-angle-side similarity. Once again, we saw SSS and SAS in our congruence postulates, but we're saying something very different here. |
Similarity postulates Similarity Geometry Khan Academy.mp3 | We're completely constraining the length of this side. The length of this side is going to have to be that same scale as that over there. We call that side-angle-side similarity. Once again, we saw SSS and SAS in our congruence postulates, but we're saying something very different here. We're saying that in SAS, if the ratio between corresponding sides of the true triangle are the same, so AB and XY of one corresponding side, and then another corresponding side, that's that second side, so that's between BC and YZ, and the angle between them are congruent, then we're saying it's similar. For SAS for congruency, we said that the sides actually had to be congruent. Here we're saying that the ratio between the corresponding sides just has to be the same. |
Similarity postulates Similarity Geometry Khan Academy.mp3 | Once again, we saw SSS and SAS in our congruence postulates, but we're saying something very different here. We're saying that in SAS, if the ratio between corresponding sides of the true triangle are the same, so AB and XY of one corresponding side, and then another corresponding side, that's that second side, so that's between BC and YZ, and the angle between them are congruent, then we're saying it's similar. For SAS for congruency, we said that the sides actually had to be congruent. Here we're saying that the ratio between the corresponding sides just has to be the same. For example, SAS, just to apply it, if I have, let me just show some examples here. Let's say I have a triangle here that is 3, 2, 4. Let's say we have another triangle here that has length 9, 6, and we also know that the angle in between are congruent, so that that angle is equal to that angle. |
Similarity postulates Similarity Geometry Khan Academy.mp3 | Here we're saying that the ratio between the corresponding sides just has to be the same. For example, SAS, just to apply it, if I have, let me just show some examples here. Let's say I have a triangle here that is 3, 2, 4. Let's say we have another triangle here that has length 9, 6, and we also know that the angle in between are congruent, so that that angle is equal to that angle. What SAS in the similarity world tells you is that these triangles are definitely going to be similar triangles, that we're actually constraining, because there's actually only one triangle we can draw right over here. It's a triangle where all of the sides are going to have to be scaled up by the same amount. There's only one long side right here that we can actually draw, and that's going to have to be scaled up by 3 as well. |
Similarity postulates Similarity Geometry Khan Academy.mp3 | Let's say we have another triangle here that has length 9, 6, and we also know that the angle in between are congruent, so that that angle is equal to that angle. What SAS in the similarity world tells you is that these triangles are definitely going to be similar triangles, that we're actually constraining, because there's actually only one triangle we can draw right over here. It's a triangle where all of the sides are going to have to be scaled up by the same amount. There's only one long side right here that we can actually draw, and that's going to have to be scaled up by 3 as well. This is the only possible triangle. If you constrain this side, you're saying, look, this is 3 times that side, this is 3 times that side, and the angle between them is congruent. There's only one triangle we can make, and we know that there is a similar triangle there, where everything is scaled up by a factor of 3, so that one triangle we can draw has to be that one similar triangle. |
Similarity postulates Similarity Geometry Khan Academy.mp3 | There's only one long side right here that we can actually draw, and that's going to have to be scaled up by 3 as well. This is the only possible triangle. If you constrain this side, you're saying, look, this is 3 times that side, this is 3 times that side, and the angle between them is congruent. There's only one triangle we can make, and we know that there is a similar triangle there, where everything is scaled up by a factor of 3, so that one triangle we can draw has to be that one similar triangle. This is what we're talking about SAS. We're not saying that this side is congruent to that side, or that side is congruent to that side. We're saying that they're scaled up by the same factor. |
Similarity postulates Similarity Geometry Khan Academy.mp3 | There's only one triangle we can make, and we know that there is a similar triangle there, where everything is scaled up by a factor of 3, so that one triangle we can draw has to be that one similar triangle. This is what we're talking about SAS. We're not saying that this side is congruent to that side, or that side is congruent to that side. We're saying that they're scaled up by the same factor. If we had another triangle that looked like this, so maybe this is 9, this is 4, and the angle between them were congruent, you couldn't say that they're similar, because this side is scaled up by a factor of 3, this side is only scaled up by a factor of 2, so this one right over there, you could not say that it is necessarily similar. Likewise, if you had a triangle that had length 9 here and length 6 there, but you did not know that these two angles are the same, once again, you're not constraining this enough, and you would not know that those two triangles are necessarily similar, because you don't know that middle angle is the same. You might be saying, well, there were a few other postulates that we had. |
Similarity postulates Similarity Geometry Khan Academy.mp3 | We're saying that they're scaled up by the same factor. If we had another triangle that looked like this, so maybe this is 9, this is 4, and the angle between them were congruent, you couldn't say that they're similar, because this side is scaled up by a factor of 3, this side is only scaled up by a factor of 2, so this one right over there, you could not say that it is necessarily similar. Likewise, if you had a triangle that had length 9 here and length 6 there, but you did not know that these two angles are the same, once again, you're not constraining this enough, and you would not know that those two triangles are necessarily similar, because you don't know that middle angle is the same. You might be saying, well, there were a few other postulates that we had. We had AAS when we dealt with congruency, but if you think about it, we've already shown that two angles by themselves are enough to show similarity, so why worry about an angle and angle and a side, or the ratio between the sides? Why even worry about that? We also had angle, side, angle, and congruence, but once again, we already know that two angles are enough, so we don't need to throw in this extra side, so we don't even need this right over here. |
Similarity postulates Similarity Geometry Khan Academy.mp3 | You might be saying, well, there were a few other postulates that we had. We had AAS when we dealt with congruency, but if you think about it, we've already shown that two angles by themselves are enough to show similarity, so why worry about an angle and angle and a side, or the ratio between the sides? Why even worry about that? We also had angle, side, angle, and congruence, but once again, we already know that two angles are enough, so we don't need to throw in this extra side, so we don't even need this right over here. So these are going to be our similarity postulates, and I want to remind you, side, side, side, this is different than the side, side, side for congruence. We're talking about the ratio between corresponding sides. We're not saying that they're actually congruent, and here, side, angle, side, it's different than the side, angle, side for congruence. |
Products and exponents raised to an exponent properties Algebra I Khan Academy.mp3 | Let's say I have two numbers, a and b, and I'm going to raise it to, I could do it in the abstract, I could raise it to the c power, but I'll do it a little bit more concrete. Let's raise it to the fourth power. What is that going to be equal to? Well, that's going to be equal to, that's going to be equal to, I could write it like this, let me copy and paste this, copy and paste, that's going to be equal to a b times a b, times a b, times a b, times a b. What is that equal to? Well, when you just multiply a bunch of numbers like this, it doesn't matter what order you're going to multiply it in. This right over here is going to be equivalent to a times a times a times a, times, we have four b's as well that we're multiplying together, times b, times b, times b, times b. |
Products and exponents raised to an exponent properties Algebra I Khan Academy.mp3 | Well, that's going to be equal to, that's going to be equal to, I could write it like this, let me copy and paste this, copy and paste, that's going to be equal to a b times a b, times a b, times a b, times a b. What is that equal to? Well, when you just multiply a bunch of numbers like this, it doesn't matter what order you're going to multiply it in. This right over here is going to be equivalent to a times a times a times a, times, we have four b's as well that we're multiplying together, times b, times b, times b, times b. What is that equal to? This right over here is a to the fourth power, and this right over here is b, b to the fourth power. You see, if you take the product of two numbers and you raise them to some exponent, that's equivalent to taking each of the numbers to that exponent and then taking their product. |
Products and exponents raised to an exponent properties Algebra I Khan Academy.mp3 | This right over here is going to be equivalent to a times a times a times a, times, we have four b's as well that we're multiplying together, times b, times b, times b, times b. What is that equal to? This right over here is a to the fourth power, and this right over here is b, b to the fourth power. You see, if you take the product of two numbers and you raise them to some exponent, that's equivalent to taking each of the numbers to that exponent and then taking their product. Here I just used the example of four, but you could do this really with any arbitrary, you could do this with actually any integer, or actually any exponent this property holds. You could satisfy yourself by trying different values and using the same logic right over here, but this is a general property that, let me write it this way, that if I have a to the b, a to the b to the c power, that this is going to be equal to a to the c, a to the c times b to the c, times b to the c, times b to the c power. We'll use this throughout actually mathematics when we try to simplify things or rewrite an expression in a different way. |
Products and exponents raised to an exponent properties Algebra I Khan Academy.mp3 | You see, if you take the product of two numbers and you raise them to some exponent, that's equivalent to taking each of the numbers to that exponent and then taking their product. Here I just used the example of four, but you could do this really with any arbitrary, you could do this with actually any integer, or actually any exponent this property holds. You could satisfy yourself by trying different values and using the same logic right over here, but this is a general property that, let me write it this way, that if I have a to the b, a to the b to the c power, that this is going to be equal to a to the c, a to the c times b to the c, times b to the c, times b to the c power. We'll use this throughout actually mathematics when we try to simplify things or rewrite an expression in a different way. Now let me introduce you another core idea here, and this is the idea of raising something to some power, and I'll just use the example of three, and then raising that to some power. What could this be simplified as? Well, let's think about it. |
Products and exponents raised to an exponent properties Algebra I Khan Academy.mp3 | We'll use this throughout actually mathematics when we try to simplify things or rewrite an expression in a different way. Now let me introduce you another core idea here, and this is the idea of raising something to some power, and I'll just use the example of three, and then raising that to some power. What could this be simplified as? Well, let's think about it. This is the same thing as a to the third, this is the same thing, let me copy and paste that, is a to the third times a to the third. What is a to the third times? This is equal to a to the third times a to the third, and that's going to be equal to a to the three plus three power. |
Products and exponents raised to an exponent properties Algebra I Khan Academy.mp3 | Well, let's think about it. This is the same thing as a to the third, this is the same thing, let me copy and paste that, is a to the third times a to the third. What is a to the third times? This is equal to a to the third times a to the third, and that's going to be equal to a to the three plus three power. We have the same base, and so we would add, and they're being multiplied, they're being raised to these two exponents, so it's going to be the sum of the exponents, which of course is going to be equal to a to the, that's a different color, a, it's going to be a to the sixth power. What just happened over here? Well, I took two a to the thirds, and I multiplied them together, so I took these two threes and added them together. |
Products and exponents raised to an exponent properties Algebra I Khan Academy.mp3 | This is equal to a to the third times a to the third, and that's going to be equal to a to the three plus three power. We have the same base, and so we would add, and they're being multiplied, they're being raised to these two exponents, so it's going to be the sum of the exponents, which of course is going to be equal to a to the, that's a different color, a, it's going to be a to the sixth power. What just happened over here? Well, I took two a to the thirds, and I multiplied them together, so I took these two threes and added them together. This essentially right over here, you could view this as two times three. This right over here is two times three. That's how we got the six. |
Products and exponents raised to an exponent properties Algebra I Khan Academy.mp3 | Well, I took two a to the thirds, and I multiplied them together, so I took these two threes and added them together. This essentially right over here, you could view this as two times three. This right over here is two times three. That's how we got the six. When I raise something to one exponent, and then raised it to another, that's equivalent to raising the base to the product of those two exponents. I just did it with this example right over here, but I encourage you, try other numbers to see how this works. I could do this in general. |
Products and exponents raised to an exponent properties Algebra I Khan Academy.mp3 | That's how we got the six. When I raise something to one exponent, and then raised it to another, that's equivalent to raising the base to the product of those two exponents. I just did it with this example right over here, but I encourage you, try other numbers to see how this works. I could do this in general. I could say a to the b power, and then let me copy and paste that. Copy, and then I'm going to raise that to the c power. I'm going to raise that to the c power. |
Products and exponents raised to an exponent properties Algebra I Khan Academy.mp3 | I could do this in general. I could say a to the b power, and then let me copy and paste that. Copy, and then I'm going to raise that to the c power. I'm going to raise that to the c power. What is that going to give me? I'm essentially going to have to take c of these, so one, two, three, I don't know how large of a number c is, so I'll just do dot, dot, dot. Dot, dot, dot. |
Products and exponents raised to an exponent properties Algebra I Khan Academy.mp3 | I'm going to raise that to the c power. What is that going to give me? I'm essentially going to have to take c of these, so one, two, three, I don't know how large of a number c is, so I'll just do dot, dot, dot. Dot, dot, dot. I have c of these right over here. I have c of these, so there's c of them right over there. What is that going to be equal to? |
Products and exponents raised to an exponent properties Algebra I Khan Academy.mp3 | Dot, dot, dot. I have c of these right over here. I have c of these, so there's c of them right over there. What is that going to be equal to? That is going to be equal to a to the, well for each of these c, I'm going to have a b that I'm going to add together. Let me write this. I'm going to have a b plus b plus b plus dot, dot, dot plus b. |
Products and exponents raised to an exponent properties Algebra I Khan Academy.mp3 | What is that going to be equal to? That is going to be equal to a to the, well for each of these c, I'm going to have a b that I'm going to add together. Let me write this. I'm going to have a b plus b plus b plus dot, dot, dot plus b. Now I have c of these b's, so I have c b's right over here, or you could view this as a, this is equal to a to the c times b power. C or a, you could view a to the c b power. Very useful. |
Products and exponents raised to an exponent properties Algebra I Khan Academy.mp3 | I'm going to have a b plus b plus b plus dot, dot, dot plus b. Now I have c of these b's, so I have c b's right over here, or you could view this as a, this is equal to a to the c times b power. C or a, you could view a to the c b power. Very useful. If someone were to say, what is 35 to the third power, and then that raised to the seventh power, well this is obviously going to be a huge number, but we can at least simplify the expression. This is going to be equal to 35 to the product of these two exponents. It's going to be 35 to the three times seven, or 35 to the 21st power. |
Positive and negative slope Algebra I Khan Academy.mp3 | So, for example, if we're looking at the XY plane here, our change in the vertical direction is gonna be a change in the Y variable divided by change in the horizontal direction is gonna be a change in the X variable. And so let's see why that is a good definition for slope. Well, I could draw something with a slope of one. A slope of one might look something like, let me, I could start it over here, and let me get my line tool out. So slope of one, as X increases by one, Y increases by one. As X increases by one, Y increases by one. So slope of one is going to look like, is going to look like this. |
Positive and negative slope Algebra I Khan Academy.mp3 | A slope of one might look something like, let me, I could start it over here, and let me get my line tool out. So slope of one, as X increases by one, Y increases by one. As X increases by one, Y increases by one. So slope of one is going to look like, is going to look like this. Is going to look like, is going to look like this. Notice, as I have a change in X, however much my change in X is, so for example here, my change in X is plus two, is positive two, I'm gonna have the same change in Y. My change in Y is going to be plus two. |
Positive and negative slope Algebra I Khan Academy.mp3 | So slope of one is going to look like, is going to look like this. Is going to look like, is going to look like this. Notice, as I have a change in X, however much my change in X is, so for example here, my change in X is plus two, is positive two, I'm gonna have the same change in Y. My change in Y is going to be plus two. So my change in Y divided by change in X is two divided by two is one. So for this line, I have a slope, slope is equal to one. But what would a slope of two look like? |
Positive and negative slope Algebra I Khan Academy.mp3 | My change in Y is going to be plus two. So my change in Y divided by change in X is two divided by two is one. So for this line, I have a slope, slope is equal to one. But what would a slope of two look like? Well, a slope of two should be steeper, and we could draw that. A slope of two, I could start at that same point. Actually, why don't I start at the same point, and we'll see, but you don't have to. |
Positive and negative slope Algebra I Khan Academy.mp3 | But what would a slope of two look like? Well, a slope of two should be steeper, and we could draw that. A slope of two, I could start at that same point. Actually, why don't I start at the same point, and we'll see, but you don't have to. Actually, let me start at a different point. So if I start over, let's say here, a slope of two would look like, a slope of two would look like, for every one that I increase in the X direction, I'm gonna increase two in the Y direction. So it's gonna look like, it is going to look like that. |
Positive and negative slope Algebra I Khan Academy.mp3 | Actually, why don't I start at the same point, and we'll see, but you don't have to. Actually, let me start at a different point. So if I start over, let's say here, a slope of two would look like, a slope of two would look like, for every one that I increase in the X direction, I'm gonna increase two in the Y direction. So it's gonna look like, it is going to look like that. This line right over here, you see it. If my change in X is one, change in X is equal to one, my change in Y, my change in Y, my change in Y is two. So change in Y over change in X is gonna be two over one. |
Positive and negative slope Algebra I Khan Academy.mp3 | So it's gonna look like, it is going to look like that. This line right over here, you see it. If my change in X is one, change in X is equal to one, my change in Y, my change in Y, my change in Y is two. So change in Y over change in X is gonna be two over one. The slope here is two. And now, hopefully, you're appreciating why this definition of slope is a good one. The higher the slope, the steeper it is. |
Positive and negative slope Algebra I Khan Academy.mp3 | So change in Y over change in X is gonna be two over one. The slope here is two. And now, hopefully, you're appreciating why this definition of slope is a good one. The higher the slope, the steeper it is. The faster it increases, the faster we increase in the vertical direction as we increase in the horizontal direction. Now, what would a negative slope be? So let's just think about what a line with a negative slope would mean. |
Positive and negative slope Algebra I Khan Academy.mp3 | The higher the slope, the steeper it is. The faster it increases, the faster we increase in the vertical direction as we increase in the horizontal direction. Now, what would a negative slope be? So let's just think about what a line with a negative slope would mean. Well, a negative slope would mean that, well, we could take an example. If we have our change in Y over change in X, was, say, let's say it was equal to a negative one. That means that if we have a change in X of one, then in order to get negative one here, that means that our change in Y would have to be equal to negative one. |
Positive and negative slope Algebra I Khan Academy.mp3 | So let's just think about what a line with a negative slope would mean. Well, a negative slope would mean that, well, we could take an example. If we have our change in Y over change in X, was, say, let's say it was equal to a negative one. That means that if we have a change in X of one, then in order to get negative one here, that means that our change in Y would have to be equal to negative one. So a line with a negative slope, a negative one slope would look like, let me see if I can draw it, would look like, a negative one slope would look like, would look like this. Notice, as X increases, as X increases by a certain amount, so our delta X here is one, Y decreases by that same amount instead of increasing. So now this is what we consider a downward sloping line. |
Positive and negative slope Algebra I Khan Academy.mp3 | That means that if we have a change in X of one, then in order to get negative one here, that means that our change in Y would have to be equal to negative one. So a line with a negative slope, a negative one slope would look like, let me see if I can draw it, would look like, a negative one slope would look like, would look like this. Notice, as X increases, as X increases by a certain amount, so our delta X here is one, Y decreases by that same amount instead of increasing. So now this is what we consider a downward sloping line. So change in Y is equal to negative one. So our change in Y over change in X, change in Y over change in X is equal to negative one over one, which is equal to negative one. So the slope of this line is negative one. |
Positive and negative slope Algebra I Khan Academy.mp3 | So now this is what we consider a downward sloping line. So change in Y is equal to negative one. So our change in Y over change in X, change in Y over change in X is equal to negative one over one, which is equal to negative one. So the slope of this line is negative one. Now, if you had a slope of negative two, it would decrease even faster. So a line with a slope of negative two, it could look something like this. Let me draw it. |
Positive and negative slope Algebra I Khan Academy.mp3 | So the slope of this line is negative one. Now, if you had a slope of negative two, it would decrease even faster. So a line with a slope of negative two, it could look something like this. Let me draw it. So as X increases by one, Y would decrease by two. As X increases by one, Y would decrease by two. So it would look something like, let me see if I can, it would look like, it would look like that. |
Positive and negative slope Algebra I Khan Academy.mp3 | Let me draw it. So as X increases by one, Y would decrease by two. As X increases by one, Y would decrease by two. So it would look something like, let me see if I can, it would look like, it would look like that. Notice, as our X increases by a certain amount, our Y decreases, decreases by twice as much. So this right over here has a slope. This has a slope of negative two. |
Positive and negative slope Algebra I Khan Academy.mp3 | So it would look something like, let me see if I can, it would look like, it would look like that. Notice, as our X increases by a certain amount, our Y decreases, decreases by twice as much. So this right over here has a slope. This has a slope of negative two. So hopefully this gives you a little bit more intuition for what slope represents and how the number that we used to represent slope, how you can use that to visualize how steep a line is. A very high positive slope, as X increases, Y is going to increase fairly dramatically. If you have a negative slope, you're actually going to go from, you're actually going to go, as X increases, your Y is actually going to decrease. |
Factoring difference of squares shared factors High School Math Khan Academy.mp3 | What binomial factor do they share? And like always, pause the video and see if you can work through this. All right, now let's work through this together. And the way I'm gonna do it is I'm just gonna try to factor both of them into the product of binomials and maybe some other things, and see if we have any common binomial factors. So first, let's focus on m squared minus four m minus 45. So let me write it over here. M squared minus four m minus 45. |
Factoring difference of squares shared factors High School Math Khan Academy.mp3 | And the way I'm gonna do it is I'm just gonna try to factor both of them into the product of binomials and maybe some other things, and see if we have any common binomial factors. So first, let's focus on m squared minus four m minus 45. So let me write it over here. M squared minus four m minus 45. So when you're factoring a quadratic expression like this, where the coefficient on the, in this case, m squared term, on the second degree term is one, we could factor it as being equal to m plus a times m plus b, where a plus b is going to be equal to this coefficient right over here, and a times b is going to be equal to this coefficient right over here. So let's be clear. So a, let me just, another color. |
Factoring difference of squares shared factors High School Math Khan Academy.mp3 | M squared minus four m minus 45. So when you're factoring a quadratic expression like this, where the coefficient on the, in this case, m squared term, on the second degree term is one, we could factor it as being equal to m plus a times m plus b, where a plus b is going to be equal to this coefficient right over here, and a times b is going to be equal to this coefficient right over here. So let's be clear. So a, let me just, another color. So a plus b needs to be equal to negative four. A plus b needs to be equal to negative four. And then a times b needs to be equal to negative 45. |
Factoring difference of squares shared factors High School Math Khan Academy.mp3 | So a, let me just, another color. So a plus b needs to be equal to negative four. A plus b needs to be equal to negative four. And then a times b needs to be equal to negative 45. A times b is equal to negative 45. Now I like to focus on the a times b and think about, well, what could a and b be to get to negative 45? Well, if I'm taking the product of two things, and if the product is negative, that means that they're going to have different signs. |
Factoring difference of squares shared factors High School Math Khan Academy.mp3 | And then a times b needs to be equal to negative 45. A times b is equal to negative 45. Now I like to focus on the a times b and think about, well, what could a and b be to get to negative 45? Well, if I'm taking the product of two things, and if the product is negative, that means that they're going to have different signs. And if when we add them, we get a negative number, that means that the negative one has a larger magnitude. So let's think about this a little bit. So a times b is equal to negative 45. |
Factoring difference of squares shared factors High School Math Khan Academy.mp3 | Well, if I'm taking the product of two things, and if the product is negative, that means that they're going to have different signs. And if when we add them, we get a negative number, that means that the negative one has a larger magnitude. So let's think about this a little bit. So a times b is equal to negative 45. So this could be, let's try some values out. So one and 45, those are too far apart. Let's see, three and 15, those still seem pretty far apart. |
Factoring difference of squares shared factors High School Math Khan Academy.mp3 | So a times b is equal to negative 45. So this could be, let's try some values out. So one and 45, those are too far apart. Let's see, three and 15, those still seem pretty far apart. Let's see, it looks like five and nine seem interesting. So if we say five times, if we were to say five times negative nine, that indeed is equal to negative 45. And five plus negative nine is indeed equal to negative four. |
Factoring difference of squares shared factors High School Math Khan Academy.mp3 | Let's see, three and 15, those still seem pretty far apart. Let's see, it looks like five and nine seem interesting. So if we say five times, if we were to say five times negative nine, that indeed is equal to negative 45. And five plus negative nine is indeed equal to negative four. So a could be equal to five, and b could be equal to negative nine. And so if we were to factor this, this is going to be m plus five times m, I could say m plus negative nine, but I'll just write m minus nine. So just like that, I've been able to factor, I've been able to factor this first quadratic expression right over there as a product of two binomials. |
Factoring difference of squares shared factors High School Math Khan Academy.mp3 | And five plus negative nine is indeed equal to negative four. So a could be equal to five, and b could be equal to negative nine. And so if we were to factor this, this is going to be m plus five times m, I could say m plus negative nine, but I'll just write m minus nine. So just like that, I've been able to factor, I've been able to factor this first quadratic expression right over there as a product of two binomials. So now let's try to factor the other quadratic expression. Let's try to factor six m squared minus 150. And let's see, the first thing I might wanna do is both six, both six m squared and 150, they're both divisible by six. |
Factoring difference of squares shared factors High School Math Khan Academy.mp3 | So just like that, I've been able to factor, I've been able to factor this first quadratic expression right over there as a product of two binomials. So now let's try to factor the other quadratic expression. Let's try to factor six m squared minus 150. And let's see, the first thing I might wanna do is both six, both six m squared and 150, they're both divisible by six. So let me write it this way. I could write it as six, actually I'll just write six m squared minus six times, let's see, six goes into 150 25 times. So all I did is I rewrote this, and really I just wrote 150 as six times 25. |
Factoring difference of squares shared factors High School Math Khan Academy.mp3 | And let's see, the first thing I might wanna do is both six, both six m squared and 150, they're both divisible by six. So let me write it this way. I could write it as six, actually I'll just write six m squared minus six times, let's see, six goes into 150 25 times. So all I did is I rewrote this, and really I just wrote 150 as six times 25. And now you can clearly see that we can factor out a six. You can view this as undistributing the six. So this is the same thing as six times m squared minus 25, which we recognize this is a difference of squares, so it's all gonna be six times m plus five times m minus five. |
Factoring difference of squares shared factors High School Math Khan Academy.mp3 | So all I did is I rewrote this, and really I just wrote 150 as six times 25. And now you can clearly see that we can factor out a six. You can view this as undistributing the six. So this is the same thing as six times m squared minus 25, which we recognize this is a difference of squares, so it's all gonna be six times m plus five times m minus five. And so we've factored this out as a product of binomials and a constant factor here, six. And so what is their shared common, or what is their common binomial factor that they share? Well you see when we factor it out, they both have an m plus five. |
Constructing, solving two-step inequality example Linear inequalities Algebra I Khan Academy.mp3 | My brain immediately says that's greater than or equal to 120,000. If they had an audience of 45,000 in Mesa and another 33,000 in Denver, how many people attended their show in Las Vegas? So let's say Las Vegas. I'll just use L for Las Vegas. So the number of people who attended their show in Las Vegas plus the number that attended their show in Mesa, which is 45,000, plus the number of people that attended their show in Denver, which is 33,000. Those are our three cities right there. Las Vegas, Mesa, and Denver. |
Constructing, solving two-step inequality example Linear inequalities Algebra I Khan Academy.mp3 | I'll just use L for Las Vegas. So the number of people who attended their show in Las Vegas plus the number that attended their show in Mesa, which is 45,000, plus the number of people that attended their show in Denver, which is 33,000. Those are our three cities right there. Las Vegas, Mesa, and Denver. That has to be at least 120,000 people. Or another way of interpreting that is greater than or equal to 120,000. So to figure out how many people attended their show in Las Vegas, we just solve for L on this inequality. |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.