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

Objective Solve quadratic equations by completing the square.

In the previous lesson, you solved quadratic equations by isolating x2 and then using square roots. This method works if the quadratic equation, when written in standard form, is a perfect square. When a trinomial is a perfect square, there is a relationship between the coefficient of the x-term and the constant term. X2 + 6x + 9 x2 – 8x + 16 Divide the coefficient of the x-term by 2, then square the result to get the constant term.

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 relationship shown above 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 – 5x + 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 + 8x + 16

Example 2A: Solving x2 +bx = c by Completing the Square Solve by completing the square. Check your answer. 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.

Check It Out! Example 2a Solve by completing the square. Check your answer. 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.

Solving a Quadratic Equation by Completing the Square