Factoring Polynomials

Holt Algebra Completing the Square 9-8 Completing the Square Holt Algebra 1 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz

Holt Algebra Completing the Square Warm Up Simplify

Holt Algebra Completing the Square Warm Up Solve each quadratic equation by factoring. 5. x 2 + 8x + 16 = 0 6. x 2 – 22x = 0 7. x 2 – 12x + 36 = 0 x = –4 x = 11 x = 6

Holt Algebra Completing the Square Solve quadratic equations by completing the square. Objective

Holt Algebra Completing the Square An expression in the form x 2 + bx is not a perfect square. However, you can use the algorithm below to add a term to x 2 + bx to form a trinomial that is a perfect square. This is called completing the square.

Holt Algebra Completing the Square Example 1: Completing the Square Complete the square to form a perfect square trinomial. A. x 2 + 2x + B. x 2 – 6x + x 2 + 2x x 2 + 2x + 1 x 2 + –6x x 2 – 6x + 9 Identify b..

Holt Algebra Completing the Square Check It Out! Example 1 Complete the square to form a perfect square trinomial. a. x x + b. x 2 – 5x + x x x x + 36 x 2 + –5x Identify b. x 2 – 6x +.

Holt Algebra Completing the Square Check It Out! Example 1 Complete the square to form a perfect square trinomial. c. 8x + x 2 + x 2 + 8x x x + 16 Identify b..

Holt Algebra Completing the Square To solve a quadratic equation in the form x 2 + bx = c, first complete the square of x 2 + bx. Then you can solve using square roots.

Holt Algebra Completing the Square Solving a Quadratic Equation by Completing the Square

Holt Algebra Completing the Square Example 2A: Solving x 2 +bx = c Solve by completing the square. x x = –15 Step 1 x x = –15 Step 2 Step 3 x x + 64 = – Step 4 (x + 8) 2 = 49 Step 5 x + 8 = ± 7 Step 6 x + 8 = 7 or x + 8 = –7 x = –1 or x = –15 The equation is in the form x 2 + bx = c. Complete the square. Factor and simplify. Take the square root of both sides. Write and solve two equations..

Holt Algebra Completing the Square Example 2B: Solving x 2 +bx = c Solve by completing the square. x 2 – 4x – 6 = 0 Step 1 x 2 + (–4x) = 6 Step 3 x 2 – 4x + 4 = Step 4 (x – 2) 2 = 10 Step 5 x – 2 = ± √10 Write in the form x 2 + bx = c. Complete the square. Factor and simplify. Take the square root of both sides. Write and solve two equations. Step 6 x – 2 = √10 or x – 2 = – √10 x = 2 + √10 or x = 2 – √ 10. Step 2

Holt Algebra Completing the Square Check It Out! Example 2a Solve by completing the square. x x = –9 Step 1 x x = –9 Step 3 x x + 25 = – Complete the square. The equation is in the form x 2 + bx = c. Step 2 Step 4 (x + 5) 2 = 16 Step 5 x + 5 = ± 4 Step 6 x + 5 = 4 or x + 5 = –4 x = –1 or x = –9 Factor and simplify. Take the square root of both sides. Write and solve two equations..

Holt Algebra Completing the Square Example 3A: Solving ax 2 + bx = c by Completing the Square Solve by completing the square. –3x x – 15 = 0 Step 1 x 2 – 4x + 5 = 0 x 2 – 4x = –5 x 2 + (–4x) = –5 Step 3x 2 – 4x + 4 = –5 + 4 Divide by – 3 to make a = 1. Write in the form x 2 + bx = c. Complete the square.. Step 2

Holt Algebra Completing the Square Example 3A Continued Solve by completing the square. –3x x – 15 = 0 Step 4 (x – 2) 2 = –1 There is no real number whose square is negative, so there are no real solutions. Factor and simplify. Step 3x 2 – 4x + 4 = –5 + 4

Holt Algebra Completing the Square Complete the square to form a perfect square trinomial. 1. x 2 +11x + 2. x 2 – 18x + Solve by completing the square. 3. x 2 – 2x – 1 = x 2 + 6x = x x = 23 Lesson Quiz: Part I 81 6, –8