A52 APPENDIX Review Discriminant of a Quadratic Equation For a quadratic equation + + = ax bx c 0: 2 • If − > b ac 4 0, 2 there are two unequal real solutions. • If − = b ac 4 0, 2 there is a repeated real solution, a root of multiplicity 2. • If − < b ac 4 0, 2 there is no real solution. THEOREM Quadratic Formula Consider the quadratic equation + + = ≠ ax bx c a 0 0 2 • If − < b ac 4 0, 2 this equation has no real solution. • If − ≥ b ac 4 0, 2 the real solution(s) of this equation is (are) given by the quadratic formula . = − ± − x b b ac a 4 2 2 The quantity − b ac 4 2 is called the discriminant of the quadratic equation, because its value tells us whether the equation has real solutions. In fact, it also tells us how many solutions to expect. When asked to find the real solutions of a quadratic equation, always evaluate the discriminant first to see if there are any real solutions. Solving a Quadratic Equation by Using the Quadratic Formula Find the real solutions, if any, of the equation − + = x x 3 5 1 0. 2 Solution EXAMPLE 8 The equation is in standard form, so we compare it to + + = ax bx c 0 2 to find a, b, and c. − + = + + = x x ax bx c 3 5 1 0 0 2 2 = =− = a b c 3, 5, 1 With = = − a b 3, 5, and = c 1, evaluate the discriminant − b ac 4 . 2 ( ) − = − − ⋅ ⋅ = − = b ac 4 5 431 25 12 13 2 2 Since − > b ac 4 0, 2 there are two real solutions. Use the quadratic formula with = = − = a b c 3, 5, 1, and − = b ac 4 13. 2 ( ) = − ± − = − − ± ⋅ = ± x b b ac a 4 2 5 13 2 3 5 13 6 2 The solution set is { } − + 5 13 6 , 5 13 6 . Now Work problem 113
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