Write each expression as a trinomial. Warm Up Write each expression as a trinomial. 1. (x – 5)2 x2 – 10x + 25 Factor each expression. 2. x2 – 18 + 81 (x – 9)2

Objectives Solve quadratic equations by completing the square.

If a quadratic expression of the form x2 + bx cannot model a square, you can add a term to form a perfect square trinomial. This is called completing the square.

Example 1: Completing the Square Complete the square for the expression. Write the resulting expression as a binomial squared. x2 – 14x + Find . x2 – 14x + 49 Add. (x – 7)2 Factor.

Example 2: Completing the Square Complete the square for the expression. Write the resulting expression as a binomial squared. x2 + 9x + Find . Add. Factor.

Example 3 Complete the square for the expression. Write the resulting expression as a binomial squared. x2 + 4x + Find . x2 + 4x + 4 Add. (x + 2)2 Factor.

Example 4 Complete the square for the expression. Write the resulting expression as a binomial squared. x2 – 4x + Find . x2 – 4x + 4 Add. (x – 2)2 Factor.

Example 5 Complete the square for the expression. n2 + 4n - 24 = 0 n2 + 4n = 24 n2 + 4n + 4 = 24 + 4 (n + 2)2 = 28    

CLASSWORK Worksheet

Example 1: Solving a Quadratic Equation by Completing the Square Solve the equation by completing the square. x2 = 12x – 20 Collect variable terms on one side. x2 – 12x = –20 Set up to complete the square. x2 – 12x + = –20 + Add to both sides. x2 – 12x + 36 = –20 + 36 Simplify.

Example 1 Continued (x – 6)2 = 16 Factor. Take the square root of both sides. Simplify. x – 6 = ±4 x – 6 = 4 or x – 6 = –4 Solve for x. x = 10 or x = 2

Example 2: Solving a Quadratic Equation by Completing the Square Solve the equation by completing the square. 18x + 3x2 = 45 x2 + 6x = 15 Divide both sides by 3. x2 + 6x + = 15 + Set up to complete the square. Add to both sides. x2 + 6x + 9 = 15 + 9 Simplify.

Example 2 Continued (x + 3)2 = 24 Factor. Take the square root of both sides. Simplify.

Example 3 Solve the equation by completing the square. x2 – 2 = 9x Collect variable terms on one side. x2 – 9x = 2 Set up to complete the square. x2 – 9x + = 2 + Add to both sides. Simplify.

Take the square root of both sides. Example 3 Continued Factor. ± 9 2 x – 89 4 = Take the square root of both sides. ± 9 2 x = 89 Simplify.

Example 4 Solve the equation by completing the square. 3x2 – 24x = 27 Divide both sides by 3. x2 – 8x = 9 Set up to complete the square. x2 –8x + = 9 + Add to both sides. Simplify.

Example 4 Continued Solve the equation by completing the square. Factor. Take the square root of both sides. Simplify. x – 4 =–5 or x – 4 = 5 Solve for x. x =–1 or x = 9

CLASSWORK Worksheet