138 CHAPTER 1 Equations and Inequalities CAUTION The restriction on a, b, and c is important. For example, x2 - 25x - 1 = 0 has discriminant b2 - 4ac = 5 + 4 = 9, which would indicate two rational solutions if the coefficients were integers. By the quadratic formula, the two solutions 25 {3 2 are irrational numbers. When the numbers a, b, and c are integers (but not necessarily otherwise), the value of the discriminant b2 - 4ac can be used to determine whether the solutions of a quadratic equation are rational, irrational, or nonreal complex numbers. The number and type of solutions based on the value of the discriminant are shown in the following table. Solutions of Quadratic Equations Discriminant Number of Solutions Type of Solutions Positive, perfect square Two Rational Positive, but not a perfect square Two Irrational Zero One (a double solution) Rational Negative Two Nonreal complex As seen in Example 5 As seen in Example 6 EXAMPLE 9 Using the Discriminant Evaluate the discriminant for each equation. Then use it to determine the number of distinct solutions, and tell whether they are rational, irrational, or nonreal complex numbers. (a) 5x2 + 2x - 4 = 0 (b) x2 - 10x = -25 (c) 2x2 - x + 1 = 0 SOLUTION (a) For 5x2 + 2x - 4 = 0, use a = 5, b = 2, and c = -4. b2 - 4ac = 22 - 41521-42 = 84 Discriminant The discriminant 84 is positive and not a perfect square, so there are two distinct irrational solutions. (b) First, write the equation in standard form as x2 - 10x + 25 = 0. Thus, a = 1, b = -10, and c = 25. b2 - 4ac = 1-1022 - 41121252 = 0 Discriminant There is one distinct rational solution, a double solution. (c) For 2x2 - x + 1 = 0, use a = 2, b = -1, and c = 1. b2 - 4ac = 1-122 - 4122112 = -7 Discriminant There are two distinct nonreal complex solutions. (They are complex conjugates.) S Now Try Exercises 83, 85, and 89.
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