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Stitz-Zeager_College_Algebra_e-book

Of symmetry x 1 y 10 9 8 7 6 5 4 3 2 1 2 1 1 x 2 y x 4

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Unformatted text preview: 4a b2 − 4ac 4a = 1 a = b2 − 4ac 4a2 2 b 2a =± √ b2 − 4ac 4a2 b2 − 4ac 2a √ b b2 − 4ac x=− ± 2a 2a √ −b ± b2 − 4ac x= 2a b x+ 2a =± extract square roots 144 Linear and Quadratic Functions In our discussions of domain, we were warned against having negative numbers underneath the √ square root. Given that b2 − 4ac is part of the Quadratic Formula, we will need to pay special attention to the radicand b2 − 4ac. It turns out that the quantity b2 − 4ac plays a critical role in determining the nature of the solutions to a quadratic equation. It is given a special name and is discussed below. Definition 2.6. If a, b, c are real numbers with a = 0, then the discriminant of the quadratic equation ax2 + bx + c = 0 is the quantity b2 − 4ac. Theorem 2.2. Discriminant Trichotomy: Let a, b, and c be real numbers with a = 0. • If b2 − 4ac < 0, the equation ax2 + bx + c = 0 has no real solutions. • If b2 − 4ac = 0, the equation ax2 + bx + c = 0 has exactly one real solution. • If b2 − 4ac > 0, the equation ax2 + bx + c = 0 has exactly two real solutions. The proof of Theorem 2.2 stems from the position of the discriminant in the quadratic equation, and is left as a good mental exercise for the reader. The next example exploits...
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