Read the explanations, follow the examples, take notes, and complete the You Trys. 1 Derive the Quadratic Formula In Section 10.1, we saw that any quadratic equation of the form ax2 bx c 0 (a 0) can be solved by completing the square. Therefore, we can solve equations like x2 8x 5 0 and 2x2 3x 1 0 using this method. We can develop another method for solving quadratic equations by completing the square on the general quadratic equation ax2 bx c 0 (a 0). This will let us derive the quadratic formula. The steps we use to complete the square on ax2 bx c 0 are exactly the same steps we use to solve an equation like 2x2 3x 1 0. We will do these steps side by side so that you can more easily understand how we are solving ax2 bx c 0 for x by completing the square. Solve for x by Completing the Square. 2x2 3x 1 0 ax2 bx c 0 Step 1: The coeffi cient of the squared term must be 1. 2x2 3x 1 0 ax2 bx c 0 2x2 2 3x 2 1 2 0 2 ax2 a Divide by 2. Divide by a. bx a c a 0 a x2 3 2 x 1 2 0 x2 b a x c a 0 Simplify. Simplify. Step 2: Get the constant on the other side of the equal sign. x2 3 2 x 1 2 Add 1 2 . x2 b a x c a Subtract c a . Step 3: Complete the square. 1 2 a3 2 b 3 4 1 2 of x-coeffi cient 1 b a2 a b b 2a 1 2 of x-coeffi cient a3 4 b 2 9 16 Square the result. a b 2a b 2 b2 4a2 Square the result. Add 9 16 to both sides of the equation. Add b2 4a2 to both sides of the equation. x2 3 2 x 9 16 1 2 9 16 x2 b a x b2 4a2 c a b2 4a2 x2 3 2 x 9 16 8 16 9 16 Get a common denominator. x2 b a x b2 4a2 4ac 4a2 b2 4a2 x2 3 2 x 9 16 17 16 Add. x2 b a x b2 4a2 Get a common denominator. b2 4ac 4a2 Add. Notice that we are using the same steps that we used in Section 10.1 to solve both equations. 624 CHAPTER 10 Quadratic Equations and Functions www.mhhe.com/messersmith
messersmith_power_intermediate_algebra_1e_ch4_7_10
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