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

Warm Up Simplify. 1. 2. 19 3. 4.

Warm Up Solve each quadratic equation by factoring. 5. x2 + 8x + 16 = 0 6. x2 – 22x + 121 = 0 7. x2 – 12x + 36 = 0 x = –4 x = 11 x = 6

Objective Solve quadratic equations by completing the square.

An expression in the form x2 + bx is not a perfect square An expression in the form x2 + bx is not a perfect square. However, you can use the algorithm below to add a term to x2 + bx to form a trinomial that is a perfect square. This is called completing the square.

Example 1: Completing the Square Complete the square to form a perfect square trinomial. A. x2 + 2x + B. x2 – 6x + x2 + 2x x2 + –6x Identify b. . x2 + 2x + 1 x2 – 6x + 9

Check It Out! Example 1 Complete the square to form a perfect square trinomial. a. x2 + 12x + b. x2 – 5x + x2 + 12x x2 + –5x Identify b. . x2 – 6x + x2 + 12x + 36

Check It Out! Example 1 Complete the square to form a perfect square trinomial. c. 8x + x2 + x2 + 8x Identify b. . x2 + 12x + 16

To solve a quadratic equation in the form x2 + bx = c, first complete the square of x2 + bx. Then you can solve using square roots.

Solving a Quadratic Equation by Completing the Square

Example 2A: Solving x2 +bx = c Solve by completing the square. x2 + 16x = –15 The equation is in the form x2 + bx = c. Step 1 x2 + 16x = –15 Step 2 . Step 3 x2 + 16x + 64 = –15 + 64 Complete the square. Step 4 (x + 8)2 = 49 Factor and simplify. Take the square root of both sides. Step 5 x + 8 = ± 7 Step 6 x + 8 = 7 or x + 8 = –7 x = –1 or x = –15 Write and solve two equations.

Example 2B: Solving x2 +bx = c Solve by completing the square. x2 – 4x – 6 = 0 Write in the form x2 + bx = c. Step 1 x2 + (–4x) = 6 Step 2 . Step 3 x2 – 4x + 4 = 6 + 4 Complete the square. Step 4 (x – 2)2 = 10 Factor and simplify. Take the square root of both sides. Step 5 x – 2 = ± √10 Step 6 x – 2 = √10 or x – 2 = –√10 x = 2 + √10 or x = 2 – √10 Write and solve two equations.

Check It Out! Example 2a Solve by completing the square. x2 + 10x = –9 The equation is in the form x2 + bx = c. Step 1 x2 + 10x = –9 Step 2 . Step 3 x2 + 10x + 25 = –9 + 25 Complete the square. Factor and simplify. Step 4 (x + 5)2 = 16 Take the square root of both sides. Step 5 x + 5 = ± 4 Step 6 x + 5 = 4 or x + 5 = –4 x = –1 or x = –9 Write and solve two equations.

Example 3A: Solving ax2 + bx = c by Completing the Square Solve by completing the square. –3x2 + 12x – 15 = 0 Step 1 Divide by – 3 to make a = 1. x2 – 4x + 5 = 0 x2 – 4x = –5 Write in the form x2 + bx = c. x2 + (–4x) = –5 Step 2 . Step 3 x2 – 4x + 4 = –5 + 4 Complete the square.

Example 3A Continued Solve by completing the square. Step 3 x2 – 4x + 4 = –5 + 4 –3x2 + 12x – 15 = 0 Step 4 (x – 2)2 = –1 Factor and simplify. There is no real number whose square is negative, so there are no real solutions.

Lesson Quiz: Part I Complete the square to form a perfect square trinomial. 1. x2 +11x + 2. x2 – 18x + Solve by completing the square. 3. x2 – 2x – 1 = 0 4. 3x2 + 6x = 144 5. 4x2 + 44x = 23 81 6, –8