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The Quadratic Formula and the Discriminant Lesson 6-4 Here is the quadratic formula-which is proved by completing the square. The Quadratic Formula __________________________________. If ax² + bx + c = 0, Then Example 1. Use the quadratic formula to solve this quadratic equation: 3x² + 5x − 8 = 0 Solution. We have: a = 3, b = 5, c = −8. Therefore, according to the formula: x = −5 + 11 6 or −5 − 11 6 6 16 or 6 6 These are the two __________ or __________________. x 1or 8 3 And they are__________ . When the roots are rational, we could have solved the equation by factoring, which is always the simplest method. 3x² + 5x − 8 x = = (3x + 8)(x −1 ) − 8 3 or 1. Example 2. Use the quadratic formula to find the roots of each quadratic. a) x² − 5x + 5 a= b = , c = b) 2x² − 8x + 5 a= , b = , c = c) 5x² − 2x + 2 a = , b = , c = Discriminant Copy the chart p. 356 onto formula sheet The radicand b² − 4ac is called the discriminant. • If the discriminant is • a) Positive: • and a perfect square 2 real, rational • and is not a perfect square 2 real, irrational • b) Negative: The roots are 2 imaginary • c) Zero: 1 real Example 3: Find the discriminant and find the nature of the roots. a. 4 x 2 25 20 x a= ,b= ,c= Describe the nature of the roots. b. 3x 2 2 5 x a= , b = , c = Describe the nature of the roots