Unformatted text preview: 4a
b2 − 4ac
4a = 1
a = b2 − 4ac
4a2 2 b
√ b2 − 4ac
4a2 b2 − 4ac
b2 − 4ac
−b ± b2 − 4ac
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
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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