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Copyright © 2013, 2009, 2005 Pearson Education, Inc. Section 8.4 Quadratic Formula

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. Objectives Solving Quadratic Equations The Discriminant Quadratic Equations Having Complex Solutions

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. The solutions to ax 2 + bx + c = 0 with a 0 are given by QUADRATIC FORMULA

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example Solve the equation 4x 2 + 3x – 8 = 0. Support your results graphically. Solution Symbolic Solution Let a = 4, b = 3 and c = 8. or

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example (cont) 4x 2 + 3x – 8 = 0 Graphical Solution

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example Solve the equation 3x 2 6x + 3 = 0. Support your result graphically. Solution Let a = 3, b = 6 and c = 3.

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example Solve the equation 2x 2 + 4x + 5 = 0. Support your result graphically. Solution Let a = 2, b = 4 and c = 5. There are no real solutions for this equation because is not a real number.

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. To determine the number of solutions to the quadratic equation ax 2 + bx + c = 0, evaluate the discriminant b 2 – 4ac. 1. If b 2 – 4ac > 0, there are two real solutions. 2. If b 2 – 4ac = 0, there is one real solution. 3. If b 2 – 4ac < 0, there are no real solutions; there are two complex solutions. THE DISCRIMINANT AND QUADRATIC EQUATIONS

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example Use the discriminant to determine the number of solutions to 2x 2 + 5x = 3. Then solve the equation using the quadratic formula. Solution 2x 2 + 5x 3 = 0 Let a = 2, b = 5 and c = 3. Thus, there are two solutions. b 2 – 4ac = (5) 2 – 4(2)(3)= 1 or

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. If k > 0, the solution to x 2 + k = 0 are given by THE EQUATION x 2 + k = 0

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example Solve x = 0. Solution The solutions are

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example Solve 3x 2 – 7x + 5 = 0. Write your answer in standard form: a + bi. Solution Let a = 3, b = 7 and c = 5. and

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example Solve Write your answer in standard form: a + bi. Solution Begin by adding 2x to each side of the equation and then multiply by 5 to clear fractions. Let a = 2, b = 10 and c = 15.

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example (cont) Let a = 2, b = 10 and c = 15.

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example Solve by completing the square. Solution After applying the distributive property, the equation becomes Since b = 4,add to each side of the equation. The solutions are 2 + i and 2 i.

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