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Lesson 9-6 Perfect Squares and Factoring

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Determine whether each trinomial is a perfect square trinomial. If so, factor it. Questions to ask. 16x 2 + 32x + 64 Is the first term a perfect square?Yes, 16x 2 = (4x) 2 Is the last term a perfect square?Yes, 64 = 8 2 Is the middle term equal to 2(4x)(8)?No, 32x 2(4x)(8) 16x 2 + 32x + 64 is not a perfect square trinomial.

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Determine whether each trinomial is a perfect square trinomial. If so, factor it. 25x 2 - 30x + 9 49x 2 + 42x + 36

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Number of TermsFactoring Technique 2 or moreGreatest Common Factor 2Difference of Squaresa 2 - b 2 = (a + b)(a - b) 3 Perfect square trinomiala 2 + 2ab + b 2 = (a + b) 2 a 2 - 2ab + b 2 = (a - b) 2 x 2 + bx + cx 2 + bx + c = (x + m)(x + n), when m + n = b and mn = c. ax 2 + bx + cax 2 + bx + c = ax 2 + mx + nx+ c, when m + n = b and mn = ac. Then using factoring by grouping 4 or moreFactoring by groupingax + bx + ay + by = x(a + b) + y(a + b) = (a + b)(x + y)

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Factor each polynomial 4x 2 - 36 First check for the GCF. Then, since the polynomial has two terms, check for the difference of two squares. 4x 2 - 36 = 4(x 2 - 9) 4 is the GCF = 4(x 2 - 3 2 )x 2 = x x, and 9 = 3 3 = 4(x + 3)(x - 3)factor the difference of squares. Ex. 2Factor Completely

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Ex. 2Factor Completely 25x 2 + 5x - 6 This polynomial has three terms that have a GCF of 1. While the first term is a perfect square 25x 2 = (5x) 2, the last term is not. Therefore, this is not a perfect square trinomial. This trinomial is one of the form ax 2 + bx + c. Are there two numbers m and n whose product is 25 -6 or -150 and whose sum is 5? Yes, the product of 15 and -10 is -150 and their sum is 5. 25x 2 + 5x - 6 = 25x 2 + mx + nx - 6 Write the pattern = 25x 2 + 15x - 10x - 6 m = 15 and n = -10 = (25x 2 + 15x) + ( -10x - 6) Group terms with common factors. = 5x(5x + 3) -2 ( 5x + 3)Factor out the GCF for each grouping. = (5x + 3) (5x - 2)5x + 3 is the common factor.

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Factor each polynomial. 6x 2 - 96 16x 2 + 8x -15

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Ex. 3Solve Equations with Repeated Factors. Solve x 2 - x + ¼ = 0 x 2 - x + ¼ = 0 Original equation x 2 - 2(x)(½) + (½) 2 = 0 Recognize x 2 - x + ¼ as a perfect square trinomial (x - ½) 2 = 0 Factor the perfect square trinomial. x - ½ = 0 Set repeated factor equal to zero. x = ½Solve for x.

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Factor each polynomial. 4x 2 + 36x + 81

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Key Concept For any number n > 0, if x 2 = n, then x = x 2 = 9 x 2 = or 3

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Ex. 4Use the Square Root Property to Solve Equations Solve (a + 4) 2 = 49 (a + 4) 2 = 49Original equation a + 4 = Square Root Property a + 4 = 7 49 = 7 7 a = -4 7 Subtract 4 from each side. a = -4 + 7 or a = -4 - 7Separate into two equations. a = 3 or a = -11Simplify

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Ex. 4Use the Square Root Property to Solve Equations Solve y 2 -4y + 4 = 25 y 2 -4y + 4 = 25 Original equation (y) 2 -2(y)(2) + 2 2 Recognize perfect square trinomials. (y - 2) 2 = 25 Factor perfect square trinomial. y - 2 = Square root property. y - 2 = 525 = 5 5 y = 2 + 5 or y = 2 - 5Separate into two equations. y = 7 or y = -3Simplify

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Ex. 4Use the Square Root Property to Solve Equations Solve (x - 3) 2 = 5 (x - 3) 2 = 5 Original equation x - 3 = Square root property. x = 3 Add 3 to each side. x = 3 + or x = 3 - Separate into two equations. x ≈ 5.24 or x ≈ 0.76Simplify

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Solve each equation. Check your solutions. (b - 7) 2 = 36 y 2 + 12y + 36 = 100 (x + 9) 2 = 8

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