4-3 Solving Quadratic Equations by Factoring. Write a quadratic equation in standard form with the given root(s) SOLUTION: Write the pattern.
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1 17. 7 Write a quadratic equation in standard form with the given root(s). Write the pattern. Since there is only one root, it is a repeated root. Replace p and q with 7. Use the FOIL method to multiply. esolutions Manual - Powered by Cognero Page 1
2 19. Write the pattern. Replace p and q with Use the FOIL method to multiply. Multiply each side by 5. Factor each polynomial. 21. The GCF of the two terms is 17c. Factor the GCF. esolutions Manual - Powered by Cognero Page 2
3 23. Factor out 3. Use the identity a 2 b 2 = (a + b)(a b) to factor x Factor 12c from the first two terms and d from the last two terms. Factor 4g + 3f from the two terms.. esolutions Manual - Powered by Cognero Page 3
4 27. Find the factors of 22 whose sum is 9. 2( 11) = 22 and 2 + ( 11) = 9 Write 9x as 2x 11x. Factor x from the first two terms and 11 from the last two terms. Factor x + 2 from the two terms. 29. Here, a = 15, b = 7 and c = 2. ac = 15( 2) = 30 Find two factors of 30 whose sum is 7. 10( 3) = 30 and 10 + ( 3) = 7 Write 7x as 10x 3x. Factor 5x from the first two terms and 1 from the last two terms. Factor 3x + 2 from the two terms. esolutions Manual - Powered by Cognero Page 4
5 31. Here, a = 18, b = 15 and c = 12. ac = 18( 12) = 216 Find two factors of 216 whose sum is ( 9) = 216 and 24 + ( 9) = 15 Write 15x as 24x + ( 9)x. Factor 6x from the first two terms and 3 from the last two terms. Factor 3x + 4 from the two terms. 33. Use the identity a 2 b 2 = (a + b)(a b) esolutions Manual - Powered by Cognero Page 5
6 35. Factor 3 from all the three terms. Factor 5x 2 28x 12. Here, a = 5, b = 28 and c = 12. ac = 5( 12) = 60 Find two factors of 60 whose sum is (2) = 60 and = 28 Write 28x as 30x + 2x. Factor 5x from the first two terms and 2 from the last two terms. Factor x 6 from the two terms. esolutions Manual - Powered by Cognero Page 6
7 37. Factor out the GCF, 12x. Use the identity a 2 b 2 = (a + b)(a b) to factor y 2 9. esolutions Manual - Powered by Cognero Page 7
8 Solve each equation by factoring. 39. Find the factors of 24 whose sum is 5. 3( 8) = 24 and 3 + ( 8) = 5 Write 5x as 3x + ( 5x). Factor x from the first two terms and 8 from the last two terms. Factor x + 3 from the two terms. Use the Zero Product Property. the roots are 3 and 8. esolutions Manual - Powered by Cognero Page 8
9 41. Write the equation with right side equal to zero. Use the identity a 2 b 2 = (a + b)(a b) to factor x 2 4. Use the Zero Product Property. the roots are 2 and 2. esolutions Manual - Powered by Cognero Page 9
10 43. Factor out 1. Now factor 8x 2 46x Here, a = 8, b = 46 and c = 30. ac = 8(30) = 240 Find two factors of 240 whose sum is ( 6) = 240 and 40 + ( 6) = 46 Write 46x as 40x + ( 6x). Factor 8x from the first two terms and 6 from the last two terms. Factor x 5 from the two terms. Use the Zero Product Property. the roots are esolutions Manual - Powered by Cognero Page 10
11 45. NUMBER THEORY Find two consecutive even integers with a product of 624. Let the numbers be 2n and 2(n + 1). Their product is 624. Here, a = 4, b = 4 and c = 624. ac = 4(624) = 2496 Find two factors of 2496 whose sum is 4. 52( 48) = 2496 and 52 + ( 48) = 4 Write 4n as 52n 48n. Factor 4n from the first two terms and 48 from the last two terms. Factor n + 13 from the two terms. Use the Zero Product Property. When n = 12, the numbers are 24 and 26. When n = 13, the numbers are 24 and 26. esolutions Manual - Powered by Cognero Page 11
12 GEOMETRY Find x and the dimensions of each rectangle. 47. The area of a rectangle of length l and width w is l w. Here, l = x + 4, w = x 2, and area = 432. Find factors of 440 whose sum is 2. 20(22) = 440 and = 2 When x = 22, the dimensions of the rectangle becomes negative. So, x = 20. The length of the rectangle is 24 ft and width is 18 ft. esolutions Manual - Powered by Cognero Page 12
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