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Completing the square

This document provides examples and instructions for solving quadratic equations by completing the square. It begins with examples of solving quadratic equations using the square root property. It then explains how to complete the square to write a quadratic expression as a perfect square trinomial. Examples are provided to demonstrate completing the square and using it to solve quadratic equations. The document ensures readers understand completing the square through check examples and objectives.

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5-4 Completing the Square 5-4 Completing the Square Warm Up Lesson Presentation Lesson Quiz Holt Algebra Holt Algebra 2 2
5-4 Completing the Square Warm Up Write each expression as a trinomial. 1. (x – 5)2 2. (3x + 5)2 x2 – 10x + 25 9x2 + 30x + 25 Factor each expression. 3. x2 – 18 + 81 (x – 9)2 4. 16x2 + 24x + 9 (4x + 3)2 Holt Algebra 2
5-4 Completing the Square Objectives Solve quadratic equations by completing the square. Holt Algebra 2
5-4 Completing the Square Vocabulary completing the square Holt Algebra 2
5-4 Completing the Square Many quadratic equations contain expressions that cannot be easily factored. For equations containing these types of expressions, you can use square roots to find roots. Holt Algebra 2
5-4 Completing the Square Reading Math Read Holt Algebra 2 as “plus or minus square root of a.”
5-4 Completing the Square Example 1A: Solving Equations by Using the Square Root Property Solve the equation. 4x2 + 11 = 59 4x2 = 48 x2 = 12 Subtract 11 from both sides. Divide both sides by 4 to isolate the square term. Take the square root of both sides. Simplify. Holt Algebra 2
5-4 Completing the Square Example 1B: Solving Equations by Using the Square Root Property Solve the equation. x2 + 12x + 36 = 28 (x + 6)2 = 28 Factor the perfect square trinomial Take the square root of both sides. Subtract 6 from both sides. Simplify. Holt Algebra 2
5-4 Completing the Square Check It Out! Example 1a Solve the equation. 4x2 – 20 = 5 4x2 = 25 x2 = 25 4 Add 20 to both sides. Divide both sides by 4 to isolate the square term. Take the square root of both sides. Simplify. Holt Algebra 2
5-4 Completing the Square Check It Out! Example 1b Solve the equation. x2 + 8x + 16 = 49 (x + 4)2 = 49 Factor the perfect square trinomial. Take the square root of both sides. x = –4 ± 49 Subtract 4 from both sides. x = –11, 3 Holt Algebra 2 Simplify.
5-4 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. Holt Algebra 2
5-4 Completing the Square Example 2A: Completing the Square Complete the square for the expression. Write the resulting expression as a binomial squared. x2 – 14x + Find . Add. x2 – 14x + 49 Factor. (x – 7)2 Check Find the square of the binomial. (x – 7)2 = (x – 7)(x – 7) = x2 – 14x + 49 Holt Algebra 2
5-4 Completing the Square Example 2B: Completing the Square Complete the square for the expression. Write the resulting expression as a binomial squared. x2 + 9x + Find Add. Factor. Holt Algebra 2 . Check Find the square of the binomial.
5-4 Completing the Square Check It Out! Example 2a Complete the square for the expression. Write the resulting expression as a binomial squared. x2 + 4x + Find x2 + 4x + 4 (x + 2)2 Check . Add. Factor. Find the square of the binomial. (x + 2)2 = (x + 2)(x + 2) = x2 + 4x + 4 Holt Algebra 2
5-4 Completing the Square Check It Out! Example 2b Complete the square for the expression. Write the resulting expression as a binomial squared. x2 – 4x + Find x2 – 4x + 4 (x – 2)2 Check Holt Algebra 2 . Add. Factor. Find the square of the binomial. (x – 2)2 = (x – 2)(x – 2) = x2 – 4x + 4
5-4 Completing the Square Check It Out! Example 2c Complete the square for the expression. Write the resulting expression as a binomial squared. x2 + 3x + Find Add. Factor. Holt Algebra 2 . Check Find the square of the binomial.
5-4 Completing the Square You can complete the square to solve quadratic equations. Holt Algebra 2
5-4 Completing the Square Example 3A: Solving a Quadratic Equation by Completing the Square Solve the equation by completing the square. x2 = 12x – 20 x – 12x = –20 Collect variable terms on one side. x2 – 12x + Set up to complete the square. 2 = –20 + Add x2 – 12x + 36 = –20 + 36 Holt Algebra 2 to both sides. Simplify.
5-4 Completing the Square Example 3A Continued (x – 6)2 = 16 Factor. Take the square root of both sides. x – 6 = ±4 x – 6 = 4 or x – 6 = –4 x = 10 or x = 2 Holt Algebra 2 Simplify. Solve for x.
5-4 Completing the Square Example 3B: Solving a Quadratic Equation by Completing the Square Solve the equation by completing the square. 18x + 3x2 = 45 x2 + 6x = 15 x2 + 6x + = 15 + Divide both sides by 3. Set up to complete the square. Add x2 + 6x + 9 = 15 + 9 Holt Algebra 2 to both sides. Simplify.
5-4 Completing the Square Example 3B Continued (x + 3)2 = 24 Factor. Take the square root of both sides. Simplify. Holt Algebra 2
5-4 Completing the Square Check It Out! Example 3a Solve the equation by completing the square. x2 – 2 = 9x Collect variable terms on one side. x2 – 9x = 2 x2 – 9x + =2+ Set up to complete the square. Add to both sides. Simplify. Holt Algebra 2
5-4 Completing the Square Check It Out! Example 3a Continued Factor. x– 9 = ± 89 4 2 x= Holt Algebra 2 9 ± 89 2 Take the square root of both sides. Simplify.
5-4 Completing the Square Check It Out! Example 3b Solve the equation by completing the square. 3x2 – 24x = 27 Divide both sides by 3. x2 – 8x = 9 x2 –8x + =9+ Set up to complete the square. Add to both sides. Simplify. Holt Algebra 2
5-4 Completing the Square Check It Out! Example 3b Continued Solve the equation by completing the square. Factor. Take the square root of both sides. Simplify. x – 4 =–5 or x – 4 = 5 x =–1 or x = 9 Holt Algebra 2 Solve for x.