This pattern will always hold true and can be helpful in factoring some perfect square trinomials. 2 Solve an Equation of the Form ax2 bx c 0 by Completing the Square Any quadratic equation of the form ax2 bx c 0 (a 0) can be written in the form (x h)2 k by completing the square. Once an equation is in this form, we can use the square root property to solve for the variable. Procedure Steps for Solving a Quadratic Equation (ax2 bx c 0) by Completing the Square Step 1: The coeffi cient of the squared term must be 1. If it is not 1, divide both sides of the equation by a to obtain a leading coeffi cient of 1. Step 2: Get the variables on one side of the equal sign and the constant on the other side. Step 3: Complete the square. Find half of the linear coeffi cient, then square the result. Add that quantity to both sides of the equation. Step 4: Factor. Step 5: Solve using the square root property. Solve by completing the square. a) x2 12x 27 0 b) k2 2k 5 0 Solution a) x2 12x 27 0 Step 1: The coeffi cient of x2 is already 1. Step 2: Get the variables on one side of the equal sign and the constant on the other side: x2 12x 27 Step 3: Complete the square: 1 2 (12) 6 62 36 Add 36 to both sides of the equation: x2 12x 36 27 36 x2 12x 36 9 Step 4: Factor: (x 6)2 9 Step 5: Solve using the square root property. (x 6)2 9 x 6 19 x 6 3 b R x 6 3 or x 6 3 x 3 or x 9 The check is left to the student. The solution set is {9, 3}. EXAMPLE 2 In-Class Example 2 Solve by completing the square. a) v2 10v 16 0 b) z2 4z 13 0 Answer: a) {8, 2} b) www.mhhe.com/messersmith SECTION 10.2 Solving Quadratic Equations by Completing the Square 615
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