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Quadratic Formula and the Discriminant Lesson 6-4

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The quadratic formula Here is the quadratic formula -- which is proved by completing the square In other words, the quadratic formula completes the square for us. Theorem. If ax² + bx + c = 0, Theorem. Then

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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:formula That is, x= −5 + 11 6 or −5 − 11 6 These are the two roots. And they are rational. When the roots are rational,rational we could have solved the equation by factoring, which is always the simplest method. 3x² + 5x − 8 = (3x + 8)(x −1 ) x = − 8383 or 1.

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Example 2. Use the quadratic formula to find the roots of each quadratic. a) x² − 5x + 5 a=1 b = −5, c = 5. b) 2x² − 8x + 5 a=2, b = -8, c = 5

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c) 5x² − 2x + 2 a = 5, b = −2, c = 2 X = 2 ± 6i 10 = 1 ± 3i 5

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Discriminant Copy the chart p. 356 onto formula sheet The radicand b² − 4ac is called the discriminant. radicand 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

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Example 3: Find the discriminant and find the nature of the roots. a. a = 4, b = -20, c = 25 Discriminant is - 4(4)(25) = 400-400 = 0 One real, rational root b. a= 3, b = -5, c = 2 Discriminant is Two real rational roots.

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