Chapter 10: Summary Definition/Procedure Example 10.1 Solving Quadratic Equations Using the Square Root Property The Square Root Property Let k be a constant. If x2 k, then x 1k or x 1k. (p. 606) Solve (t 8)2 5. t 8 15 Square root property t 8 15 Add 8 to both sides. The solution set is {8 15, 8 15}. 10.2 Solving Quadratic Equations by Completing the Square A perfect square trinomial is a trinomial whose factored form is the square of a binomial. (p. 613) To fi nd the constant needed to complete the square for x2 bx: Step 1: Find half of the coeffi cient of x: 1 2 b Step 2: Square the result: a1 2 2 bb Step 3: Add it to x2 bx: x2 bx a1 2 (p. 613) 2 bb 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. (p. 615) Perfect Square Trinomial Factored Form g2 6g 9 (g 3)2 4t2 20t 25 (2t 5)2 Complete the square for x2 10x to obtain a perfect square trinomial. Then, factor. Step 1: Find half of the coeffi cient of x: 1 2 (10) 5 Step 2: Square the result: 52 25 Step 3: Add 25 to x2 10x: x2 10x 25 The perfect square trinomial is x2 10x 25. The factored form is (x 5)2. Solve x2 8x 9 0 by completing the square. x2 8x 9 0 The coeffi cient of x2 is 1. x2 8x 9 Get the constant on the other side Complete the square. 1 2 (8) 4 (4)2 16 Add 16 to both sides of the equation. x2 8x 16 9 16 (x 4)2 7 Factor. x 4 17 Square root property x 4 17 The solution set is {4 17, 4 17}. 10.3 Solving Quadratic Equations Using the Quadratic Formula The Quadratic Formula The solutions of any quadratic equation of the form ax2 bx c 0 (a 0) are x b 2b2 4ac 2a This is called the quadratic formula. (p. 622) of the equal sign. Solve 2x2 9x 8 0 using the quadratic formula. a 2 b 9 c 8 Substitute the values into the quadratic formula. x (9) 2(9)2 4(2)(8) 2(2) x 9 181 64 4 9 117 4 The solution set is e 9 117 4 , 9 117 4 f . 654 CHAPTER 10 Quadratic Equations www.mhhe.com/messersmith
messersmith_power_introductory_algebra_1e_ch4_7_10
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