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CHAPTER 9 Quadratic Equations Slide 2Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. 9.1Introduction to Quadratic Equations 9.2Solving Quadratic Equations by Completing the Square 9.3The Quadratic Formula 9.4Formulas 9.5Applications and Problem Solving 9.6Graphs of Quadratic Equations 9.7Functions

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OBJECTIVES 9.2 Solving Quadratic Equations by Completing the Square Slide 3Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc. aSolve quadratic equations of the type ax 2 = p. bSolve quadratic equations of the type (x + c) 2 = d cSolve quadratic equations by completing the square. dSolve certain applied problems involving quadratic equations of the type ax 2 = p.

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The equation x 2 = d has two real solutions when d > 0. The solutions are The equation x 2 = d has no real-number solution when d < 0. The equation x 2 = 0 has 0 as its only solution. 9.2 Solving Quadratic Equations by Completing the Square The Principle of Square Roots Slide 4Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

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EXAMPLE Solution We use the principle of square roots: x 2 = 25 x = 5 or x = –5 We check mentally that 5 2 = 25 and (–5) 2 = 25. The solutions are 5 and –5. 9.2 Solving Quadratic Equations by Completing the Square a Solve quadratic equations of the type ax 2 = p. ASolve: x 2 = 25 Slide 5Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

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EXAMPLE Solution a) Check: x 2 = 19 x 2 = 19 19 19 19 = 19 19 = 19 The solutions are 9.2 Solving Quadratic Equations by Completing the Square a Solve quadratic equations of the type ax 2 = p. B Solve: a) x 2 = 19 b) 9x 2 = 27 c) 5x 2 60 = 0 (continued) Slide 6Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

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EXAMPLE Solution b) c) 9.2 Solving Quadratic Equations by Completing the Square a Solve quadratic equations of the type ax 2 = p. A Solve: a) x 2 = 19 b) 9x 2 = 27 c) 5x 2 60 = 0 Slide 7Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

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EXAMPLE Solution a) x – 3 = 4 or x – 3 = –4 x = 7 or x = –1 The solutions are 7 and –1. We leave the check to the student. 9.2 Solving Quadratic Equations by Completing the Square b Solve quadratic equations of the type (x + c) 2 = d CSolve: a) (x – 3) 2 = 16b) (x + 3) 2 = 5 (continued) Slide 8Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

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EXAMPLE b) (x + 3) 2 = 5 The solutions check and can be written as (read as “negative three plus or minus the square root of five”). 9.2 Solving Quadratic Equations by Completing the Square b Solve quadratic equations of the type (x + c) 2 = d CSolve: a) (x – 3) 2 = 16b) (x + 3) 2 = 5 Slide 9Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

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EXAMPLE Solution x 2 + 8x + 16 = 17 (x + 4) 2 = 17 Sometimes we can factor an equation to express it as a square of a binomial. 9.2 Solving Quadratic Equations by Completing the Square b Solve quadratic equations of the type (x + c) 2 = d DSolve: x 2 + 8x + 16 = 17 Slide 10Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

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To complete the square for an expression like x 2 + bx, we take half of the coefficient of x and square it. Then we add that number, which is (b/2) 2. 9.2 Solving Quadratic Equations by Completing the Square Completing the Square Slide 11Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

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EXAMPLE Solution The coefficient of x is –16. Half of –16 is –8 and (–8) 2 = 64 Thus, x 2 – 16 becomes a perfect-square trinomial when 64 is added: x 2 – 16x + 64 is the square of x – 8 The number 64 completes the square. Check: (x – 8) 2 = (x – 8)(x – 8) = x 2 – 8x – 8x + 64 = x 2 – 16x + 64 9.2 Solving Quadratic Equations by Completing the Square c Solve quadratic equations by completing the square. EComplete the square: x 2 – 16x Slide 12Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

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EXAMPLE Solution Take half of 8 and square it to get 16. We add 16 to both sides of the equation. x 2 8x + 16 = 15 + 16 (x - 4) (x - 4) = 1 (x 4) 2 = 1 x 4 = 1 or x 4 = 1 x = 5 or x = 3 9.2 Solving Quadratic Equations by Completing the Square c Solve quadratic equations by completing the square. FSolve by completing the square. x 2 8x = 15 Slide 13Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

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EXAMPLE Solution We have x 2 14x 7 = 0 x 2 14x = 7 x 2 14x + 49 = 7 + 49 (x 7) 2 = 56 9.2 Solving Quadratic Equations by Completing the Square c Solve quadratic equations by completing the square. GSolve by completing the square. x 2 14x 7 = 0 Slide 14Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

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EXAMPLE 9.2 Solving Quadratic Equations by Completing the Square c Solve quadratic equations by completing the square. HSolve by completing the square. 3x 2 + 7x + 1 = 0 Slide 15Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

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EXAMPLE Solution The coefficient of the x 2 term must be 1. When it is not, we must multiply or divide on both sides to find an equivalent equation with an x 2 coefficient of 1. 9.2 Solving Quadratic Equations by Completing the Square c Solve quadratic equations by completing the square. HSolve by completing the square. (continued) Slide 16Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

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EXAMPLE 9.2 Solving Quadratic Equations by Completing the Square c Solve quadratic equations by completing the square. HSolve by completing the square. Slide 17Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

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To solve a quadratic equation ax 2 + bx + c = 0 by completing the square: 1. If a ≠ 1, multiply by 1/a so that the x 2 –coefficient is 1. 2. If the x 2 –coefficient is 1, add so that the equation is in the form x 2 + bx = –c, or if step (1) has been applied. 3. Take half of the x-coefficient and square it. Add the result on both sides of the equation. 4. Express the side with the variables as the square of a binomial. 9.2 Solving Quadratic Equations by Completing the Square Solving by Completing the Square (continued) Slide 18Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

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To solve a quadratic equation ax 2 + bx + c = 0 by completing the square: 5. Use the principle of square roots and complete the solution. 9.2 Solving Quadratic Equations by Completing the Square Solving by Completing the Square Slide 19Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

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EXAMPLE 1. Familiarize. A formula that fits this situation is s = 16t 2, where s is the distance, in feet, traveled by a body falling freely from rest in t seconds. We know that s is 1670 feet. 2. Translate. We know the distance is 1670 feet and that we need to solve for t. 1670 = 16t 2 9.2 Solving Quadratic Equations by Completing the Square d Solve certain applied problems involving quadratic equations of the type ax 2 = p. IThe Taipei 101 tower in Taiwan is 1670 feet tall. How long would it take an object to fall to the ground from the top? (continued) Slide 20Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

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EXAMPLE 3. Solve. 1670 = 16t 2 9.2 Solving Quadratic Equations by Completing the Square d Solve certain applied problems involving quadratic equations of the type ax 2 = p. I (continued) Slide 21Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

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EXAMPLE 4. Check. The number –10.2 cannot be a solution because time cannot be negative. s = 16(10.2) 2 = 16(104.04) = 1664.64 This answer is close. 5. State. It takes about 10.2 seconds for an object to fall to the ground from the top of Taipei 101. 9.2 Solving Quadratic Equations by Completing the Square d Solve certain applied problems involving quadratic equations of the type ax 2 = p. I Slide 22Copyright 2011, 2007, 2003, 1999 Pearson Education, Inc.

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