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triangle has area and height dimensions, given that they are binomial factors of the area. Find the base and height when x = 10 and b = 2, if the height is greater than the length. ound Up ound Up RRRRRound Up ound Up ound Up ound Up ound Up RRRRRound Up ound Up ound Up If you see any quadratic expression in the form ... |
.4.1 on special binomial products. rinomials rinomials ect Square e e e e TTTTTrinomials ect Squar ect Squar PPPPPerferferferferfect Squar rinomials rinomials ect Squar ect Square e e e e TTTTTrinomials PPPPPerferferferferfect Squar rinomials rinomials ect Squar ect Squar rinomials ect Squar rinomials Perfect square tr... |
οΏ½οΏ½ x2 + 2xy + y2 = (x + y)2 Sometimes you need to factor each term in the expression to get it into the correct form. Section 6.8 Section 6.8 Section 6.8 β More on Quadratics Section 6.8 Section 6.8 327327327327327 Example Example Example Example Example 22222 Factor 4x2 β 12xy + 9y2. Solution Solution Solution Solutio... |
β 56pmn + 16pn2) ft2 10. A = (81px2y2 β 90pxyz + 25pz2) ft2 8. A = (25py2 + 30py + 9p) ft2 The volume, V, of each cylinder below is the product of the height, p, and the radius squared. Find the radius in each case: 11. V = (98x2p + 84xp + 18p) cm3, height = 2 cm 12. V = (147pa2 β 84pab + 12pb2) m3, height = 3 m 13. V... |
finding hniques inc or all or all actor f actor f a common f a common f actor for all a common factor f or all a common f or all actor f a common f ynomial, ynomial, ms in a pol ms in a pol terterms in a pol terter ynomial, ms in a polynomial, ter ynomial, ms in a pol recognizing the difference of two squares, and rec... |
s another common factor to factor out: (3 + 5t) Using the distributive property: y(3 + 5t) β 2k(3 + 5t) = (y β 2k)(3 + 5t) Example Example Example Example Example 22222 Factor completely 8rt β 6ckt + 3ckm β 4rm. Solution Solution Solution Solution Solution Rearrange the expression and group in parentheses: 8rt β 6ckt +... |
y2 11. 12a2 + 9ab β 28ab β 21b2 13. 4a2 β 6ab + 6ab β 9b2 2. 6x2 + 9x + 4x + 6 4. 2x2 + 5x + 6x + 15 6. 6x2 + 8x + 15x + 20 8. 3c2 β c + 6c β 2 10. 3m2 + 3mn β mn β n2 12. 2x3 + 2x2y + 3xy2 + 3y3 14. 4b2 β 20bx β 2xb + 10x2 Find a value of? so that the expression will factor into two binomials. 15. 20n2 β 25n +?n β 20 ... |
PPascalβ ascalβs triang s triang ascalβ PPPPPascalβ s trianglelelelele s triang s triang ascalβ ascalβ ascalβs triang s triang ascalβ Pascalβs triangle was originally developed by the ancient Chinese. However, the French mathematician Blaise Pascal was the first person to discover the importance of all the patterns it ... |
und Up Although it just looks like a funny pile of shapes and numbers, there are a lot of real-life problems that can be solved using the patterns in Pascalβs Triangle. estigaaaaationtiontiontiontion β Pascalβs Triangle 331331331331331 estigestig estig pter 6 Invvvvvestig pter 6 In ChaChaChaChaChapter 6 In pter 6 In pt... |
= 12, and c = β320. Example (iii) above is a quadratic in y, while the others are quadratics in x. Guided Practice The general form of a quadratic equation is ax2 + bx + c = 0, where a, b, and c are numbers. Identify a, b, and c in these equations. 1. βx2 + 5x β 6 = 0 3. 4x2 β 12x + 9 = 0 5. βx2 β 4x β 4 = 0 7. 6y2 + ... |
5 3 and x = 3 14. 4x2 β 12x + 9 = 0 for x = 1 and x = 2 2 15. 16x2 β 8x + 1 = 0 for x = β 1 4 and x = 1 16. βx2 β 4x β 4 = 0 for x = 2 and x = β2 17. 64x2 + 48x + 9 = 0 for x = β 3 5 3 4 8 and x = 3 8 Zero Property β if xy = 0, then x = 0 or y = 0 (or Both) One way to solve a quadratic equation is to factor it and the... |
x β 2 = 0 27. 10x2 β 27x + 5 = 0 29. 2x2 β x β 28 = 0 Using Factoring to Solve Quadratic Equations 1) First arrange the terms in the quadratic equation so that you have zero on one side. 2) Then factor the nonzero expression (if possible). 3) Once done, you can use the zero property to find the solutions. Example 4 So... |
+ 28y + 20 = 5 β 6y2 38. 4(x2 β 5x) = β25 40. x(x + 4) + 9 = 5 42. 6x(3x β 4) β 7 = β15 44. 7x(7x + 2) + 4x3 + 3 = 2(2x3 + 1) 45. (2x + 9)2(x + 3)(x + 1)β1(x + 3)β1(x + 1) = 0 46. 2x(3x + 3) + 4(x + 1) = 1 + 2x + 2x2 31. x2 β 7x β 18 = 0 33. 4m2 + 4m β 15 = 0 35. 15k2 + 28k = β5 37. 4x2 + 6x + 1 = 3x2 + 8x 39. 3x(3x +... |
(2y + 7) 17. 2x(2x β 5) = 3(2x β 5) 12. x2 β 9x + 8 = 0 14. x(2x + 3) = 5(2x + 3) 16. (x + 2)(x β 2) = 3x 18. 2x(3x β 1) + 7 = 7(2 β 3x) 19. The product of two consecutive positive numbers is 30. Find the numbers. 20. The product of two consecutive positive odd numbers is 35. Find the numbers. 21. The area of a rectang... |
about squares and square roots. The Square Root Method If you square the numbers m and βm, you get the same answer, since m2 = (βm)2 (= p, say). If you take the square root of p, there are two possible answers, m or βm. In other words, if m2 = p, then m = Β± p. Example 1 uses the above property to find the two possible... |
x2 = 4 13. 3a2 = 75 11. x2 = 9 14. a2 = 81 12. 2x2 = 32 15. 5x2 = 180 16. Use the zero product property and factoring to verify your answers to Exercises 10β15 above. More Square Root Examples Example 3 Find the solution set of 3x2 β 7 = 101. Solution Here, you can get x2 on its own on one side of the equation, with n... |
Guided Practice Find the square roots of the expressions below. 17. c2 + 6c + 9 20. 9x2 β 24x + 16 23. 49 + 28y + 4y2 18. x2 + 14x + 49 21. 9x2 + 30x + 25 24. 4x2 + 4bx + b2 19. x2 β 6x + 9 22. 25 β 30k + 9k2 25. k2 β 8kx + 16x2 Solve the following equations by using the square root method. 26. k2 = 1 29. m2 = 432 32.... |
x β 16) cm long. Find the value of x that would give a square with an area of 108 cmΒ². 10. The product of the number of CDs that Donna and Keisha have is 16a2 + 56a + 49. If both have the same number of CDs, find how many CDs Donna has, in terms of a. Round Up Round Up Donβt forget that square roots result in two possi... |
trinomial. Solution To do this you have to add a number to the original expression. First look at the form of perfect square trinomials, and compare the coefficient of x with the constant term (the number not followed by x or x2): x2 + 2dx + d 2 = (x + d)2 The coefficient of x is 2d, while the constant term is d 2. So... |
β β 12 2 β 2 βββ β = (β6)2 = 36. So 36 must be added (giving y2 β 12y + 36). Section 7.2 β Completing the Square 343 Donβt forget: Remember the negative sign in front of the 10 coefficient. Example 4 Suppose x2 β 10x + c is a perfect square trinomial, and is equal to (x + k)2. What are the values of c and k? Solution H... |
to make each statement true. 25. x2 β 6x + c = (x + k)2 27. 4x2 + 12x + c = (2x + k)2 29. 4a2 β 4ab + cb2 = (2a + kb)2 26. x2 + 16x + c = (x + k)2 28. 9x2 + 30x + c = (3x + k)2 30. 9a2 β 12ab + cb2 = (3a + kb)2 344 Section 7.2 β Completing the Square If the Coefficient of x2 isnβt 1, Add a Number With an expression of... |
οΏ½β = β β βββ β 1 2 100 4 β βββ = β 25 2 Section 7.2 β Completing the Square 345 Check it out: + 2x x+ 10 2 can also be 25 2 β βββ β β 2 βββ. β 5 2 written as 2 x + Guided Practice The quadratics below are of the form a(x + d)2. Find the value of m and d in each equation. 31. 5x2 + 10x + m = 5(x + d)2 33. 2x2 β 28x + m ... |
x2 + 4xy Find the value of m and d in each of the following. 11. 5x2 β 40x + m = 5(x + d)2 13. 3x2 β 6x + m = 3(x + d)2 15. 4x2 + 24x + m = 4(x + d)2 12. 2x2 + 20x + m = 2(x + d)2 14. 3x2 β 30x + m = 3(x + d)2 16. 7x2 β 28x + m = 7(x + d)2 Add a term to convert each of the following into an expression of the form a(x ... |
οΏ½οΏ½β β perfect square trinomial by adding β 2 βββ to it. β b 2 Example 1 Convert x2 + 4x to a perfect square trinomial. Solution Here b = 4, so to convert this to a perfect square trinomial, you add β βββ β 4 2 β 2 βββ = =. 2 4 β 2 So x2 + 4x + 4 = (x + 2)2. Here x2 + 4x + 4 is a perfect square trinomial. Another way to... |
x 2 β βββ + = + β bx x 3) Add c to both sides: x 2 + + = + c bx x β βββ β β 2 βββ β β β βββ β β 2 βββ β b 2 b 2 β 2 βββ β β β βββ β β 2 βββ + β b 2 b 2 c Check it out: Compare this to (x + k)2 + m, and you see that β b= β βββ β 2 b= 2 and m + c. β 2 βββ β k Example 3 Express x2 + 4x + 1 in the form (x + k)2 + m. Solut... |
be expressed: x2 β 6x = (x β 3)2 β 9 2) Add c (= 3) to both sides of this equation to get: x2 β 6x + 3 = (x β 3)2 β 6 348 Section 7.2 β Completing the Square Guided Practice Express the following in the form (x + k)2 + m [or a(x + k)2 + m]. 10. x2 + 4x + 8 12. x2 + 8x + 5 14. x2 + 3x + 5 16. x2 + 4x + 5 18. x2 β 20x +... |
οΏ½οΏ½ + β¦ 3 2 5 2 Section 7.2 β Completing the Square 349 Check it out: Now you can deal with the expression in parentheses exactly as before. Check it out: This is exactly the same process as in Examples 3 and 4 on the previous page. Check it out: Completing the square can be used to find the highest point of an objectβs... |
5x + 10 10. 5x2 + 20x + 4 12. 3x2 + 42x + 49 13. A square has an area of x2 + 14x + k. Find the value of k. 14. A circle has an area of px2 + 18px + kp. Find the value of k. Round Up Round Up Thereβs been a lot of build-up to actually solving quadratic equations using the completing the square method β but itβs coming... |
to do here is to divide the equation by 4 first. Then you need to solve: x2 β 9 4 x + 1 2 = 0 1) x2 β x2 β 9 4 9 4 x = β 1 2 Move the constant to the other side x + 2β β 9 ββββ = β βββ β β 8 1 2 + 2β β 9 ββββ = βββ β β 8 49 64 Add a number to both sides to get a perfect square trinomial xββ βββ β 2 β ββββ β 9 8 = 49 6... |
this stage, but itβs easier if the coefficient of x2 is 1. Example 3 Solve 5x2 β 2x β 3 = 3x2 β 6x + 13. Solution Rearrange the equation first so that all like terms are combined: 5x2 β 2x β 3 = 3x2 β 6x + 13 2x2 + 4x β 16 = 0 Now divide through by 2 so that the coefficient of x2 is 1. x2 + 2x β 8 = 0 Take the constan... |
1) + 216 26. 5y(y β 6) + 51 = 131 27. 7b2 + 14b β 32 = 5b2 + 26b 28. 3(x2 β 5) β 9 = 111 β 12x 29. 8(x + 6)2 β 128 = 6(x + 6)2 30. 4(x + 5)2 β 200 = 2(x + 5)2 Independent Practice Solve by completing the square. 1. x2 + 2x β 3 = 0 3. a2 + 8a β 9 = 0 2. y2 β 4y β 12 = 0 4. b2 + 10b β 24 = 0 5. x2 β 12x + 20 = 0 6. 3x2 ... |
use the quadratic formula to solve quadratic equations β and youβll derive the quadratic formula itself. Key words: quadratic formula completing the square You can also use the quadratic formula to solve quadratic equations. It works every time. Quadratic Equations can be in Any Variable The standard form for a quadra... |
4 b 2 a = β β b or x ac β ac β2 b 2 a Examples Using the Quadratic Formula Example 2 Solve x2 β 5x β 14 = 0 using the quadratic formula. Solution Start by writing down the values of a, b, and c: a = 1, b = β5, and c = β14 Now very carefully substitute these values into the quadratic formula ac 14 ) Β± 5 = x + 56 = 25 2... |
: Leave these answers as radical expressions unless youβre told otherwise. Guided Practice Use the quadratic formula to solve each of the following equations. 1. x2 β 2x β 143 = 0 3. x2 + 2x β 1 = 0 5. 2x2 β 5x + 2 = 0 7. 2x2 β 7x β 3 = 0 9. 18x2 + 3x β 1 = 0 2. 2x2 + 3x β 1 = 0 4. x2 + 3x + 1 = 0 6. 3x2 β 2x β 3 = 0 8... |
+ 4 = 0 23. x2 β x β 12 = 0 25. 6x2 + 29x = 5 22. 4y2 β 9 = 0 24. 2x2 β 3x β 9 = 0 26. 7x2 + 41x = 6 27. The length of a rectangle is 20 cm more than 4 times its width. If the rectangle has an area of 75 cm2, find its dimensions. 28. The equation h = β14t2 + 12t + 2 gives the height of a tennis ball t seconds after be... |
from the question in the form of an equation: x(x + 9) = 220 This is a quadratic equation, so rearrange it to the form ax2 + bx + c = 0. x(x + 9) = 220 x2 + 9x = 220 x2 + 9x β 220 = 0 Write down a, b, and c: a = 1, b = 9, and c = β220 Now you can use the quadratic formula ac 2 b 2 a 2 ) ( 9 220 β Β± + 81 880 2 20 or x ... |
is equal to eight times the number. Find the number. 4. The sum of the squares of two consecutive odd integers is 74. Find the numbers. 5. The sum of the squares of two consecutive even integers is 340. Find the possible numbers. 6. The length of a rectangular field is 10 meters less than four times its width. Find th... |
solutions of the general quadratic equation ax2 + bx + c = 0 is equal to β b a product of the roots is, and that the. c a Round Up Round Up Quadratic equations pop up a lot in Algebra I. If you know the quadratic formula then youβll always be able to solve them by just substituting the values into the formula. Section... |
steeper (narrower) parabola. For example, the graph of y = 3x2 is steeper than the graph of y = x2. The basic shape of all quadratic graphs (that is, for any quadratic function y = ax2 + bx + c) is very similar to the ones above. Theyβre all concave up or concave down depending on the sign of a (concave up if a > 0 an... |
all. This is because x2 + 1 = 0 does not have any real solutions. So the graph of a quadratic function may cross the x-axis twice (y = x2 β 4), may touch the x-axis in one place (y = x2), or may never cross it (y = x2 + 1). It all depends on how many roots the quadratic function has. However, the graph will always hav... |
the x-axis in zero, one, or two points. What it means for you: Youβll graph quadratic functions by finding their roots. Key words: quadratic parabola intercept vertex line of symmetry root Drawing Graphs of Drawing Graphs of Quadratic Functions Quadratic Functions In this Topic youβll use methods for finding the inter... |
this time, so the graph is concave down. (i) To find the x-intercepts of the graph of y = β2x2 + 6x β 4, you need to solve: β2x2 + 6x β 4 = 0 This quadratic factors to give β2(x β 1)(x β 2) = 0. So using the zero property, x = 1 or x = 2. This means the x-intercepts are at (1, 0) and (2, 0). (ii) Put x = 0 into y = β2... |
if any), ii) Find the yβintercepts (if any), iii) Find the vertex, iv) Using the vertex, x-intercepts, and y-intercepts, graph the quadratic. 1. y = x2 β 2x 3. y = β4x2 β 4x + 3 5. y = x2 + 4x + 4 7. y = β9x2 β 6x + 3 2. y = x2 + 2x β 3 4. y = x2 β 4 6. y = βx2 + 4x + 5 Describe the characteristics of quadratic graphs ... |
one, or two points. What it means for you: Youβll graph quadratic functions by first completing the square of the equation. Key words: quadratic completing the square parabola intercept vertex line of symmetry root Quadratic Graphs and Quadratic Graphs and Completing the Square Completing the Square If there are no x-... |
2 + 1 is 0 + 1 = 1. This minimum value occurs at x = 3 (the value for x where (x β 3)2 = 0). So the coordinates of the vertex of the parabola are (3, 1). As before, the line of symmetry passes through the vertex β so the line of symmetry is x = 3. The graph of the quadratic function y = (x + k)2 + p has its vertex at (... |
-intercept β this is at y = 11. Example 3 Write 4x β x2 β 7 in the form a(x + k)2 + m, and sketch the graph. Solution 4x β x2 β 7 = βx2 + 4x β 7 Factor out β1 to make completing the square easier. = β[x2 β 4x + 7] = β[(x β 2)2 + 3] = β(x β 2)2 β 3 But (x β 2)2 is never negative β the minimum value it takes is 0. So β(x... |
seconds did the ball reach its maximum height? 2. What was the ballβs maximum height above sea level? The first 8 seconds in the flight of a paper airplane can be modeled by the quadratic h = 1 8 t2 β t + 4, where h is the height in feet and t is the time in seconds. Use this information to answer Exercises 3β4. 3. In... |
= x 4 ac β2 b 2 a The quadratic formula can be used to help draw the graph of a quadratic function y = ax2 + bx + c. By finding where y = 0 (that is, by solving ax2 + bx + c = 0), you can find the x-intercepts of the parabola. But itβs sometimes impossible to get an answer from the quadratic formula. When b2 β 4ac is ... |
[4 Γ 2 Γ (β2)] = 9 β (β16) = 9 + 16 = 25 Since b2 β 4ac is positive, the equation 2x2 + 3x β 2 = 0 has two distinct (unequal) real solutions. This in turn means that the function y = 2x2 + 3x β 2 has two real roots β its graph crosses the x-axis in two places. To work out the actual values of the roots, use the quadra... |
Practice Describe the nature of the solutions of each quadratic equation, and find the values of the solutions. 1. x2 + x β 12 = 0 3. x2 + 5x + 4 = 0 5. 2x2 + 5x + 2 = 0 7. 3x2 + 7x β 6 = 0 9. 2(5x2 + 1) = 9x 11. x(10x + 7) = β1 2. x2 + 2x β 3 = 0 4. 3x2 β 7x + 4 = 0 6. x2 + 3x β 1 = 0 8. 2x2 + 9x = 5 10. 6(2x2 + 1) +... |
ratic formula: β 4 ac 1 1 2 3 4 5 -1 x So the graph of y = x2 β 6x + 9 touches the x-axis at (3, 0). Guided Practice Describe the nature of the solutions of the quadratic equations, and find the value(s) of the solution(s). 17. x2 + 4x + 4 = 0 19. x2 + 6x = β9 21. 5x(5x β 6) = β9 23. 4x2 = β5(4x + 5) 25. 8x(x + 3) + 18... |
(ab) = β2 Β² + 4 β 5 y x -1 6 5 4 3 2 1 -1 -2 -3 -4 -5 -6 -7 So the graph of y = β2x2 + 4x β 5 never intersects the x-axis either. But this time, since a < 0, the graph is concave down (n-shaped) and stays below the x-axis. Guided Practice Use the discriminant to verify that there are no real number solutions for the q... |
β 2x + 3 = 0 has no real roots, and so the graph of y = x2 β 2x + 3 does not intersect the x-axis. Example 7 Find the values of k for which y = 5x2 β 3x + k has a double root. Solution As always, itβs a good idea to begin by writing down your a, b, and c: a = 5, b = β3, c = k So b2 β 4ac = (β3)2 β 4 Γ 5 Γ k = 9 β 20k ... |
both solutions. 17. Find the possible values of p if y = px2 β 7x β 7 has no real roots. State the number of times that the graphs of the following quadratic functions intercept the x-axis: 18. y = x2 β 3x β 28 20. y = 4x2 + 2x + 1 22. y = 5x2 + 3x + 1 19. y = 4x2 + 4x + 1 21. y = 2x2 β x β 1 23. Find all possible val... |
equation t β 2 = 0 or t β 3 = 0 Solve using the zero property t = 2 or t = 3 So the stone is at a height of 96 feet after 2 seconds (on the way up), and again after 3 seconds (on the way down). Example 2 Use the same information from Example 1. After how many seconds does the stone hit the ground? Explain your answer.... |
time β at t = 2.5 s. x 1 2 3 4 5 So the maximum height is 100 feet (which is reached at t = 5 2 = 2.5 s). Guided Practice 1. In a Physics experiment, a ball is thrown into the air from an initial height of 24 meters. Its height h (in meters) at any time t (in seconds) is given by h = β5t2 + 10t + 24. Find the maximum ... |
0 (when it was thrown) and t = 3 (when it lands). So the t-intercept at t = 3 represents the point when the ball lands. The t-intercept at t = β1 doesnβt have any real-life significance here. To find the maximum height, you need to find the vertex of the parabola β so complete the square: β16t2 + 32t + 48 = β16[t2 β 2... |
a ball directly upwards. The height of each ball above the pool in feet, h, is plotted against the time in seconds, t, since it was thrown. 11. The height of Jamesβs ball can be calculated using the equation h = β16tΒ² + 30t + 10. From what height above the pool does James throw his ball? 12. The height of Meiβs ball c... |
5x + 15 = 0 x2 β 15x + 50 = 0 (x β 10)(x β 5) = 0 x = 10 or x = 5 Divide through by 0.3 Solve using the zero property This means that the restaurant can employ either 5 people or 10 people and make a profit of $15,000. Section 7.6 β Motion Tasks and Other Applications 383 Example 2 Use the same information from Example... |
+ 4.5x = β0.3(x2 β 15x) = β0.3x(x β 15) = 0 at x = 0 and x = 15 So the graph looks like this 20 P (7.5, 16.875) 15 10 10 11 12 13 14 15 16 Number of employees x 384 Section 7.6 β Motion Tasks and Other Applications Check it out: Work out the profit like this: (β0.3 Γ 72) + (4.5 Γ 7) = (β0.3 Γ 82) + (4.5 Γ 8) = 16.8 Ex... |
maximum value of the stock? 5. When did the stock reach the maximum value? 6. When did the stock become worthless? The value, V, of Juanβs investment portfolio can be modeled by the equation V = 16t2 β 256t + 16,000, where t is the time in months. 7. What was the original value of Juanβs portfolio? 8. What was the min... |
were 3107 handshakes in total. How many people were at the reunion before the math teacher arrived and how many of the people present had the math teacher taught? Is your answer the only one possible? Explain your reasoning. 2) Look back at the original handshake problem. Change the problem in some way and investigate... |
the form of a fraction β that means it has a numerator and a denominator. Examples of rational expressions are. Rational expressions are written in the form p q, where q Ο 0. An Expression is Undefined if the Denominator is Zero If the denominator is equal to zero, then the expression is said to be undefined (see Topi... |
is any real number. Show that Jane is incorrect. Round Up Round Up This Topic about the limitations on rational numbers will help you when youβre dealing with fractions in later Topics. In Topic 8.1.2 youβll simplify rational expressions to their lowest terms. Section 8.1 β Rational Expressions 389 Topic 8.1.2 Califor... |
factoring both the numerator and denominator and then canceling the common factors β that means making sure its numerator and denominator have no common factors other than 1. For example: 66 78 1 β
6 11 = β
1 6 13 = 11 13 Example 2 Reduce the expression 56 64x to its lowest terms. Solution The greatest common factor (... |
can simplify the rational expression Simplify the following rational expressions. 14. 4 2 k β β k 16 15. 2 β 2 m c β 2 mc c 16 20 β 2 3 + k k 26 β β 3 13 k 10 )( m m m )( + 5 ) + 5 ) 17. 18. 2 m 2 k + β 6 k mk + β 2 mk m 2 2 19. Independent Practice 1. Simplify 2 2 k 2 c 2 β + ck 3 ck. Simplify ( ). Reduce each of the... |
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.2 Section 8.2 β Multiplying and Dividing Rational Expressions Section 8.2 Section 8.2 Example Example Example Example Example 22222 Multiply and simplify Solution Solution Solution Solution Solution Step 1: Factor the numerators and denominators if possible ( ) 33 2 2 a a )( 3 2 a )( β 2 a a )( ( β β
β 2 1 )( ( β + + ... |
( 3 )( + + )( 6 3 (( ) + ) )( β 3 )( a + 6 ) β 3 ) ax or 18 Guided Practice Multiply and simplify the rational expressions. 7 20 8 12 3 2 x x + + xy 3 + β 2 xy. 2 β x y β β xy ) Independent Practice Multiply and simplify the rational expressions ab + β x + β 2 x 2 + ab 5 + 2 ab 7 + a 6 + a 3 2 3. 5. 2 a 2 b 2 + β ab 2 ... |
boboboboblems b y using these y using these lems b lems b y using these lems by using these y using these lems b hniques..... hniques hniques tectectectectechniques hniques What it means for you: Youβll divide rational expressions by factoring and canceling. Key words: rational reciprocal common factor viding viding Di... |
by the reciprocal of that fraction. Dividing anything by a rational expression is the same as multiplying by the reciprocal of that expression. So you can always rewrite an expression a Γ· b in the form a (where b is any nonzero expression). β
=1 b a b Section 8.2 Section 8.2 Section 8.2 β Multiplying and Dividing Rati... |
essions essions Expr Expr vide Long Strings of vide Long Strings of ou Can Di YYYYYou Can Di ou Can Di At Once essions At Once Expressions vide Long Strings of Expr ou Can Divide Long Strings of At Once essions Expr vide Long Strings of ou Can Di Just like multiplication, you can divide any number of rational expressi... |
pq 3 q 2 Γ 2 p pq β + β pq 3 pq + 2 q. Solution Solution Solution Solution Solution Rewrite any divisions as multiplications by reciprocals. = 2 p 2 p 2 + β pq 2 q β β 2 pq 3 q 2 Γ 2 p pq β + pq pq + 2 q Factor all numerators and denominators. β + 2 q p )( 3 q p )( q p )( + )( q ) Cancel any common factors between the... |
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ound Up ound Up Itβs really important that you can justify your work step by step, because division of rational expressions can involve lots of calculations that look quite similar. 402402402402402 Section 8.2 Section 8.2 Section 8.2 β Multiplying and Dividing Rational Expressions Section 8.2 Section 8.2 Topic 8.3.1 C... |
Subtract the numerators and divide by the common denominator )( Guided Practice Simplify each expression. 7. 3 + + 12 10. 2 4 b 9 Independent Practice Simplify each expression. 1 + + 11 x + + 2 4 x 28 3 2 35 9 x 2 3 + + 18 13 + + 4 17 12 β 12 6. β βββ β β 1 2 x β 2 x 4 1 β β ββββ βββ β β β βββββ β β ββββ β 7 β β ββββ ... |
1 Add 2 3 1 +. 5 Solution Multiply the denominators to get a common denominator (3β5). Convert each fraction into an equivalent fraction with the common denominator and respective numerators 2β5 and 1β3 10 = + 15 3 15 Once you have two fractions with the same denominator, add the numerators and divide by the common de... |
included once in the LCM. Using the LCM as your common denominator makes the problem as simple as possible. Adding or Subtracting Fractions with Different Denominators When adding or subtracting fractions: 1) Find the least common multiple of the denominators. 2) Convert each fraction into an equivalent fraction with ... |
highest power of 2 is 22 = 4. The highest power of (x β 1) is (x β 1). Highest power of (x + 3) is (x + 3). So the LCM is 4(x β 1)(x + 3). Example 4 continued Step 2: Convert each fraction to an equivalent fraction with 4(x β 1)(x + 3) as the denominator β ( 10 x β )( β )( )( )( + x 1 ) + = + 20 x ( xβ x )( 1 1 ) + ( ... |
Adding and Subtracting Rational Expressions 411 Topic 8.4.1 Section 8.4 Solving Fractional Equations Solving Fractional Equations California Standards: 13.0: Students add, subtract, multiply, and divide rational expressions and functions. Students solve both computationally and conceptually challenging problems by usi... |
3) + (8x + 8) = 48x β 16 9x2 + 14x + 5 = 48x β 16 9x2 β 34x + 21 = 0 (9x β 7)(x β 3) = 0 x = 7 9 or x = 3 Guided Practice Check it out: This is a quadratic equation, so you could use the quadratic formula (see Topic 7.3.1) to work out the solutions instead of factoring. Check it out: Make sure that these solutions are... |
. Step 1: Find the least common multiple of the denominators: 2x + 1 and x β 1. The LCM for these denominators is (2x + 1)(x β 1). Step 2: Now, multiply both sides of the equation by (2x + 1)(x β 1) to eliminate the denominators from the rational expressions β )( )( 1 1 = β β
( 1 2 x + )( 1 x β ) 1 414 Section 8.4 β So... |
two numbers. 15 11. 10. One integer is 5 less than another. Find the numbers if the sum of their reciprocals is 17 66. 11. The denominator of a fraction is 3 more than its numerator. If the sum of the fraction and its reciprocal is 29 10, find the fraction. 12. The denominator of a fraction is 2 more than the numerato... |
Natasha would travel if there were no wind. (Assume that she would travel at a constant speed without a wind.) Solution Step 1 β Write the equation. The time Natasha takes to complete the race, can be written as: Timethere + Timeback = 50 minutes Using this, an equation for Natashaβs race time can be written in terms ... |
of candy for a total of $1.26. She kept four boxes and sold the rest for a total of $1.40. If she sold each box for 3 cents more than it cost her, how many boxes did she buy? 4. A teacher spent $8.40 on sets of chapter tests. If each set of tests had been 2 cents less, the teacher would have gotten two extra sets for ... |
How many fruits did JosΓ© order to begin with? 7. A wholesaler bought a batch of T-shirts for $77.00. She gave two of the T-shirts to her daughters and then sold the rest for a total of $90. If the wholesaler sold each T-shirt for $2 more than it cost her, how much did she pay for each T-shirt? Round Up Round Up The on... |
inate and the second number represents the y-coordinate. y This point represents the ordered pair (β2, 1). x 1 2 3 3 2 1 β1 β1 β2 β3 β3 β2 A relation is any set of ordered pairs. Relations are represented using set notation, and can be named using a letter: for example: m = {(1, 4), (2, 8), (3, 12), (4, 16)}. Every rel... |
g, h)} 5. f(x) = {(β1, 0), (βb, d), (e, 3), (7, βf)} 6. f(x) = {(a, βa), (b, βb), (βc, c), ( 1 2, βj)} Mapping Diagrams Can Be Used to Represent Relations One way to visualize a relation is to use a mapping diagram. In the diagram, the area on the left represents the domain, while the area on the right represents the r... |
and 8. A hollow circle would mean that the value was not included, and an arrowhead would mean that the domain continued to infinity in that direction. So, for example: y domain = {β2 < x < 8} 8 6 4 2 β2 2 4 6 8 β2 y β domain = {β 8 6 βx < < } 4 2 β2 2 4 6 8 β2 x x 422 Section 8.5 β Relations and Functions Independent... |
ll find out what functions are, and youβll say whether particular relations are functions. Key words: function relation ordered pair domain range Functions Functions A function is a special type of relation. Functions Map from the Domain to the Range A relation is any set of ordered pairs β without restriction. A funct... |
a) b) c) 2 5 7 11 a b c k n 2 5 6 7 11 a b c k n 2 5 6 7 11 a b c n Domain Range Domain Range Domain Range Solution a) and b) are NOT functions, since 7 is mapped to two different values β (7, a) and (7, c) have the same x-value. c) IS a function. Each member of the domain only maps onto one member of the range. Guide... |
(2, 6)} 12. g = {(β2, 5), (0, 1), (1, 2), (2, 5)} 13. h = {(β5, β4.5), (β3, β2.5), (1, 1.5), (3, 3.5), (5, 5.5)} 14. f = {(β3, β27), (β2, β8), (β1, β1), (0, 0), (1, 1), (2, 8)} 15. g = {(β2, 8), (0, 0), (1, 2), (2, 8), (3, 18)} 16. h = {(β3, 17), (β1, 1), (0, β1), (1, 1), (2, 7)} The Vertical Line Test Shows if a Grap... |
, 2} to generate sets of ordered pairs. Use them to determine whether the relation is a function or not. 2. m = {(x, x2 β 4)} 3. t = {(x, x + 2)} 4. k = {(x, y = (x β 2)(x + 2)} 5. p = {(x, y = Β± 4 2β x } 6. j = {(x, y = 2x β 1)} 7. b = {(x, y = x Β± (3x β 4)} 8. In the equation x2 + y2 = 9, is y a function of x? Explai... |
: Students determine the domain of independent variables and the range of dependent variables defined by a graph, a set of ordered pairs, or a symbolic expression. What it means for you: Youβll see some different ways of representing functions. Key words: function domain range Check it out: The notation m(x) is read as... |
range of P(x) when the domain of P(x) is the set {β2, β1, 0, 1, 2}. Solution The range is the set of all values of P(x) for which x Ε {β2, β1, 0, 1, 2}. So, range = {P(β2), P(β1), P(0), P(1), P(2)} = {(β2)3 + 1, (β1)3 + 1, (0)3 + 1, (1)3 + 1, (2)3 + 1} = {β7, 0, 1, 2, 9} Guided Practice For Exercises 1β6, let f (x) = ... |
itβs likely to be a function. 430 Section 8.5 β Relations and Functions Topic 8.5.4 California Standards: 16.0: Students understand the concepts of a relation and a function, determine whether a given relation defines a function, and give pertinent information about given relations and functions. 17.0: Students determ... |
are equal. b) The range of m = range of b = {β3, β4, 0, 5} Section 8.5 β Relations and Functions 431 Example 2 The domain of both of the following functions is the set of all real numbers. p = {(x, y = x3 + 3x2 + 3x + 1)} q = {(x, y = (x + 1)3} Determine whether the two functions are equal. Solution Both functions hav... |
on x. b) Find v(β3). Solution a) The function is undefined when its denominator (x β 1) is 0, so x ΟΟΟΟΟ 1. b) Substitute x = β3: 432 Section 8.5 β Relations and Functions 1 = + 9 1 β 4 10 = β = β 4 5 2 2 v(β3) = )β + 3 ( β β 3 1 So, v(β3) = β 5 2 Example 4 Determine the range of the function represented by the graph ... |
Guided Practice In Exercises 7β10, find the values in terms of h (and x, where appropriate). 7. Supposing f (x) = x2 β 2, find f (h + 2) β f (2). 8. If f (x) = 2x2 + 4x β 6, find f (h β 3) + f (2h). ( ) f x ( f x 9. Supposing f (x) = x2, find + β. 10. If m(x) = 2x β 3, find the value of Determine the domain for the fu... |
and if the domains are also equal, the functions will be equal. 434 Section 8.5 β Relations and Functions Chapter 8 Investigation ming Functions ming Functions ansforororororming Functions ansf ansf TTTTTrrrrransf ming Functions ming Functions ansf ansforororororming Functions TTTTTrrrrransf ming Functions ming Functi... |
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