Property The Discriminant and Solutions The expression under the radical, b2 4ac, is called the discriminant. The discriminant tells us what kind of solution a quadratic equation has. If a, b, and c are integers, then 1) if b2 4ac is positive and the square of an integer, the equation has two rational solutions. 2) if b2 4ac is positive but not a perfect square, the equation has two irrational solutions. 3) if b2 4ac is negative, the equation has two nonreal, complex solutions of the form a bi and a bi. 4) if b2 4ac 0, the equation has one rational solution. EXAMPLE 3 Find the value of the discriminant. Then, determine the number and type of solutions of each equation. a) z2 6z 4 0 b) 5h2 6h 2 Solution a) Is z2 6z 4 0 in the form ax2 bx c 0? Yes. Identify a, b, and c. a 1 b 6 c 4 Discriminant b2 4ac (6)2 4(1)(4) 36 16 52 Because 52 is positive but not a perfect square, the equation will have two irrational solutions. (152, or 2113, will appear in the solution, and 2113 is irrational.) b) Is 5h2 6h 2 in the form ax2 bx c 0? No. Rewrite the equation in that form, and identify a, b, and c. 5h2 6h 2 0 a 5 b 6 c 2 Discriminant b2 4ac (6)2 4(5)(2) 36 40 4 The discriminant is 4, so the equation will have two nonreal, complex solutions of the form a bi and a bi, where b 0. The discriminant is b2 4ac not 2b2 4ac. 628 CHAPTER 10 Quadratic Equations and Functions www.mhhe.com/messersmith
messersmith_power_intermediate_algebra_1e_ch4_7_10
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