Algebra-Trigonometry-CynthiaYoung-3ed16.pdf - c01a.qxd 5:34 PM Page 121 1.3 Quadratic Equations EXAMPLE 7 121 Completing the Square Solve the quadratic

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1.3 Quadratic Equations 121 EXAMPLE 8 Completing the Square When the Leading Coefficient Is Not Equal to 1 Solve the equation 3 x 2 12 x 13 0 by completing the square. Solution: Divide by the leading coefficient, 3. Collect variables to one side of the equation and constants to the other side. Add to both sides. Write the left side of the equation as a perfect square and simplify the right side. Solve using the square root method. Simplify. Rationalize the denominator (Chapter 0). Simplify. , The solution set is . YOUR TURN Solve the equation 2 x 2 4 x 3 0 by completing the square. e 2 - i 1 3 3 , 2 + i 1 3 3 f x = 2 + i 1 3 3 x = 2 - i 1 3 3 x = 2 ; i 1 3 # 1 3 1 3 x = 2 ; i A 1 3 x - 2 = ; A - 1 3 ( x - 2) 2 = - 1 3 x 2 - 4 x + 4 = - 13 3 + 4 A - 4 2 B 2 = 4 x 2 - 4 x = - 13 3 x 2 - 4 x + 13 3 = 0 EXAMPLE 7 Completing the Square Solve the quadratic equation x 2 8 x 3 0 by completing the square. Solution: Add 3 to both sides. x 2 8 x 3 Add to both sides. x 2 8 x 4 2 3 4 2 Write the left side as a perfect square and simplify the right side. ( x 4) 2 19 Apply the square root method to solve. Subtract 4 from both sides. The solution set is 4 , 4 . In Example 7, the leading coefficient (the coefficient of the x 2 term) is 1. When the leading coefficient is not 1, start by first dividing the equation by that leading coefficient. F 1 19 1 19 E x = - 4 ; 1 19 x + 4 = ; 1 19 A 1 2 # 8 B 2 = 4 2 Technology Tip Graph y 1 x 2 8 x 3. Study Tip When the leading coefficient is not 1, start by first dividing the equation by that leading coefficient. Technology Tip Graph y 1 3 x 2 12 x 13. Answer: The solution is . The solution set is e 1 - i 1 2 2 , 1 + i 1 2 2 f . x = 1 ; i 2 2 2 The x -intercepts are the solutions to this equation. The graph does not cross the x -axis, so there is no real solution to this equation.
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122 CHAPTER 1 Equations and Inequalities Study Tip Read as “negative b plus or minus the square root of the quantity b squared minus 4 ac all over 2 a .” x = - b ; 2 b 2 - 4 ac 2 a If ax 2 bx c 0, , then the solution is Note: The quadratic equation must be in standard form ( ax 2 bx c 0) in order to identify the parameters: a —coefficient of x 2 b —coefficient of x c —constant x = - b ; 2 b 2 - 4 ac 2 a a Z 0 Q UADRATIC FORMULA We read this formula as negative b plus or minus the square root of the quantity b squared minus 4ac all over 2a. It is important to note that negative b could be positive (if b is negative). For this reason, an alternate form is “opposite b . . .” The Quadratic Formula should be memorized and used when simpler methods (factoring and the square root method) cannot be used. The Quadratic Formula works for any quadratic equation. Study Tip The Quadratic Formula works for any quadratic equation. Quadratic Formula Let us now consider the most general quadratic equation: We can solve this equation by completing the square.
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  • Summer '17
  • juan alberto
  • Quadratic Formula, Quadratic equation, Equations and Inequalities

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