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OBJECTIVES © 2010 Pearson Education, Inc. All rights reserved 1 Quadratic Equations Solve a quadratic equation by factoring. Solve a quadratic equation by the square root method. Solve a quadratic equation by completing the square. Solve a quadratic equation by using the quadratic formula. Solve a quadratic equations with complex solutions. Solve applied problems. SECTION 1.4 1 2 3 4 5 6

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QUADRATIC EQUATION A quadratic equation in the variable x is an equation equivalent to the equation where a, b, and c are real numbers and a ≠ 0. © 2010 Pearson Education, Inc. All rights reserved 2

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THE ZERO-PRODUCT PROPERTY Let A and B be two algebraic expressions. Then AB = 0 if and only if A = 0 or B = 0. © 2010 Pearson Education, Inc. All rights reserved 3

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EXAMPLE 1 Page 128 # 22 © 2010 Pearson Education, Inc. All rights reserved 4

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EXAMPLE 2 Page 128 # 26 © 2010 Pearson Education, Inc. All rights reserved 5

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Suppose u is any algebraic expression and d ≥ 0. THE SQUARE ROOT PROPERTY © 2010 Pearson Education, Inc. All rights reserved 6

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EXAMPLE 3 Page 128 # 38, # 44 and # 46 © 2010 Pearson Education, Inc. All rights reserved 7

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A quadratic trinomial x in with coefficient of x 2 equal to 1 is a perfect-square trinomial if the constant term is the square of one-half the coefficient of x. PERFECT SQUARE TRNOMIAL © 2010 Pearson Education, Inc. All rights reserved 8

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EXAMPLE 4 Solving a Quadratic Equation by Completing the Square © 2010 Pearson Education, Inc. All rights reserved 9

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Step 1Rearrange the quadratic equation so that the terms in x 2 and x are on the left side of the equation and the constant term is on the right side. Step 2Make the coefficient of x 2 equal to 1 by dividing both sides of the equation by the original coefficient. (Steps 1and 2 are interchangeable.) METHOD OF COMPLETING THE SQUARE © 2010 Pearson Education, Inc. All rights reserved 10

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Step 3Add the square of one-half the coefficient of x to both sides of the equation. Step 4Write the equation in the form (x + k) 2 = d using the fact that the left side is a perfect square. METHOD OF COMPLETING THE SQUARE Step 5Take the square root of each side, prefixing ± to the right side. Step 6Solve the two equations from Step 5. © 2010 Pearson Education, Inc. All rights reserved 11

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EXAMPLE 5 Page 129 # 66 © 2010 Pearson Education, Inc. All rights reserved 12

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The solutions of the quadratic equation in the standard form ax 2 + bx + c = 0 with a ≠ 0 are given by the formula THE QUADRATIC FORMULA © 2010 Pearson Education, Inc. All rights reserved 13

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EXAMPLE 6 Page 129 # 76 and # 86 © 2010 Pearson Education, Inc. All rights reserved 14

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In the quadratic formula THE DISCRIMINANT the quantity b 2 – 4ac under the radical sign is called the discriminant of the equation. The discriminant reveals the type of solutions of the equation. © 2010 Pearson Education, Inc. All rights reserved 15

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THE DISCRIMINANT DiscriminantSolutions b 2 – 4ac > 0Two unequal real b 2 – 4ac = 0One real b 2 – 4ac < 0Two nonreal complex © 2010 Pearson Education, Inc. All rights reserved 16

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EXAMPLE 7 Using the Discriminant Use the discriminant to determine the number and type of solutions of each quadratic equation. © 2010 Pearson Education, Inc. All rights reserved 17

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