The expression under the radical, b2 4ac, in the quadratic formula is called the discriminant. Examples 1a) and 2a) show that if the discriminant is positive but not a perfect square, then the given equation has two irrational solutions. We see in Example 1b) that if the discriminant is positive and the square of an integer, then the equation has two rational solutions and can be solved by factoring. If the discriminant is negative, as in Example 2b), then the equation has no real number solution. What if the discriminant equals 0? What does that tell us about the solution set? How could knowing a little bit about the discriminant help you check your answer? Solve 2 9 k2 2 3 k 1 2 0 using the quadratic formula. Solution Is 2 9 k2 2 3 k 1 2 0 in the form ax2 bx c 0? Yes. However, working with fractions in the quadratic formula would be diffi cult. Eliminate the fractions by multiplying the equation by 18, the least common denominator of the fractions. 18 a2 9 k2 2 3 k 1 2 b 18 0 Multiply by 18 to eliminate the fractions. 4k2 12k 9 0 Identify a, b, and c: a 4 b 12 c 9 k b 2b2 4ac 2a Quadratic formula k (12) 2(12)2 4(4)(9) 2(4) Substitute a 4, b 12, and c 9. k 12 1144 144 8 Perform the operations. k 12 10 8 Simplify the radicand. The discriminant 0. k 12 0 8 12 8 3 2 The solution set is e 3 2 f . Example 3 illustrates that when the discriminant equals 0, the equation has one rational solution. EXAMPLE 3 In-Class Example 3 Solve 2 15 w2 2 3 w 5 6 0 using the quadratic formula. Answer: e 5 2 f YOU TRY 3 Solve 3 4 h2 1 2 h 1 12 0 using the quadratic formula. SECTION 10.3 www.mhhe.com/messersmith Solving Quadratic Equations Using the Quadratic Formula 625
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