Chapter 10: Summary Definition/Procedure Example 10.1 The Square Root Property and Completing the Square The Square Root Property Let k be a constant. If x2 k, then x 1k or x 1k. (p. 612) The Distance Formula The distance, d, between two points with coordinates (x1, y1) and (x2, y2) is given by d 2(x2 x1)2 (y2 y1)2. (p. 616) A perfect square trinomial is a trinomial whose factored form is the square of a binomial. (p. 616) Complete the Square for x2 bx 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 . The factored 2 bb form is ax 2 . (p. 616) 1 2 bb 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. (p. 619) Solve 6p2 54. p2 9 Divide by 6. p 19 Square root property p 3 19 3 The solution set is {3, 3}. Find the distance between the points (6, 2) and (0, 2). x1 y1 x2 y2 Label the points: ( 6 , 2 ) (0, 2 ) Substitute the values into the distance formula. d 2(0 6)2 (2 (2))2 2(6)2 (4)2 136 16 152 2113 Perfect Square Trinomial Factored Form y2 8y 16 (y 4)2 9t2 30t 25 (3t 5)2 Complete the square for x2 12x to obtain a perfect square trinomial. Then, factor. Step 1: Find half of the coeffi cient of x: 1 2 (12)6 Step 2: Square the result: 62 36 Step 3: Add 36 to x2 12x: x2 12x 36 The perfect square trinomial is x2 12x 36. The factored form is (x 6)2. Solve x2 6x 7 0 by completing the square. x2 6x 7 0 The coeffi cient of x2 is 1. x2 6x 7 Get the constant on the other side of the equal sign. 1 2 (6) 3 Complete the square: (3)2 9 Add 9 to both sides of the equation. x2 6x 9 7 9 (x 3)2 2 Factor. x 3 12 Square root property x 3 12 The solution set is {3 12, 3 12}. 692 CHAPTER 10 Quadratic Equations and Functions www.mhhe.com/messersmith
messersmith_power_intermediate_algebra_1e_ch4_7_10
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