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Perfect Square Trinomials and Factoring

Determine whether each trinomial is a perfect square trinomial and factor it. Solve equations with repeated factors using the square root property. Learn how to factor perfect square trinomials.

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Perfect Square Trinomials and Factoring

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  1. Lesson 9-6 Perfect Squares and Factoring

  2. Determine whether each trinomial is a perfect square trinomial. If so, factor it. • Questions to ask. • 16x2 + 32x + 64 • Is the first term a perfect square? Yes, 16x2 = (4x)2 • Is the last term a perfect square? Yes, 64 = 82 • Is the middle term equal to 2(4x)(8)? No, 32x  2(4x)(8) • 16x2 + 32x + 64 is not a perfect square trinomial.

  3. Determine whether each trinomial is a perfect square trinomial. If so, factor it. • 25x2 - 30x + 9 • 49x2 + 42x + 36

  4. Ex. 2 Factor Completely • Factor each polynomial • 4x2 - 36 • First check for the GCF. Then, since the polynomial has two terms, check for the difference of two squares. • 4x2 - 36 = 4(x2- 9) 4 is the GCF • = 4(x2 - 32) x2 = x  x, and 9 = 3 3 • = 4(x + 3)(x - 3) factor the difference of squares.

  5. Ex. 2 Factor Completely • 25x2 + 5x - 6 • This polynomial has three terms that have a GCF of 1. While the first term is a perfect square 25x2 = (5x)2, the last term is not. Therefore, this is not a perfect square trinomial. • This trinomial is one of the form ax2 + 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. • 25x2 + 5x - 6 • = 25x2 + mx + nx - 6 Write the pattern • = 25x2 + 15x -10x - 6 m = 15 and n = -10 • = (25x2 + 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.

  6. Factor each polynomial. • 6x2 - 96 • 16x2 + 8x -15

  7. Ex. 3 Solve Equations with Repeated Factors. • Solve x2 - x + ¼ = 0 • x2 - x + ¼ = 0 Original equation • x2 - 2(x)(½) + (½)2 = 0 Recognize x2 - x + ¼ as a perfect square trinomial • (x - ½)2 = 0 Factor the perfect square trinomial. • x - ½ = 0Set repeated factor equal to zero. • x = ½ Solve for x.

  8. Factor each polynomial. • 4x2 + 36x + 81

  9. Key Concept • For any number n > 0, if x2 = n, then x =  • x2 = 9 • x2 =  or  3

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

  11. Ex. 4 Use the Square Root Property to Solve Equations Solve y2 -4y + 4 = 25 y2 -4y + 4 = 25 Original equation (y)2 -2(y)(2) + 22Recognize perfect square trinomials. (y - 2)2 = 25 Factor perfect square trinomial. y - 2 =  Square root property. y - 2 =  5 25 = 5  5 y = 2 + 5 or y = 2 - 5 Separate into two equations. y = 7 or y = -3 Simplify

  12. Ex. 4 Use 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 ≈ 0.76 Simplify

  13. Solve each equation. Check your solutions. • (b - 7)2 = 36 • y2 + 12y + 36 = 100 • (x + 9)2 = 8

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