Discriminant

The discriminant of a quadratic equation of the form $a{x}^2+b{x}+{c}=0$ is the quantity $b^2-4ac$ . When ${a},{b},{c}$ are real, this is a notable quantity, because if the discriminant is positive, the equation has two real roots; if the discriminant is negative, the equation has two nonreal roots; and if the discriminant is 0, the equation has a real double root.

Example Problems

Introductory

(AMC 12 2005) There are two values of $a$ for which the equation $4x^2+ax+8x+9=0$ has only one solution for $x$ . What is the sum of these values of $a$ ?

Solution: Since we want the $a$ 's where there is only one solution for $x$ , the discriminant has to be $0$ . $(a+8)^2-4\times4\times9=a^2+16a-80=0$ . The sum of these values of $a$ is $-16$ .

Intermediate

1977 Canadian MO Problem 1

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Discriminant

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Example Problems

Introductory

Intermediate

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