5 Solve a Quadratic Equation 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 Solve 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. EXAMPLE 7 Solve by completing the square. a) x2 6x 8 0 b) 12h 4h2 24 Solution a) x2 6x 8 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 6x 8 1 2 (6) 3 Step 3: Complete the square: 32 9 Add 9 to both sides of the equation: x2 6x 9 8 9 x2 6x 9 1 Step 4: Factor: (x 3)2 1 Step 5: Solve using the square root property. (x 3)2 1 x 3 11 x 3 1 b R x 3 1 or x 3 1 x 2 or x 4 The check is left to the student. The solution set is 54, 26. Are you writing out the steps as you are reading them? www.mhhe.com/messersmith SECTION 10.1 The Square Root Property and Completing the Square 619
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