Discriminant

Revision as of 21:53, 21 April 2008 by I like pie (talk | contribs)

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

Other resources