Difference between revisions of "Discriminant"

Revision as of 13:47, 14 May 2014

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 $0$ . $(a+8)^2-4(4)(9)=a^2+16a-80=0$ . The sum of these values of $a$ is $-16$ .

Intermediate

1977 Canadian MO Problem 1

Other resources

Wikipedia entry

@@ Line 5: / Line 5: @@
 * (AMC 12 2005) There are two values of <math>a</math> for which the equation <math>4x^2+ax+8x+9=0</math> has only one solution for <math>x</math>. What is the sum of these values of <math>a</math>?
-Solution: Since we want the <math>a</math>'s where there is only one solution for <math>x</math>, the discriminant has to be <math>0</math>. <math>(a+8)^2-4\times4\times9=a^2+16a-80=0</math>. The sum of these values of <math>a</math> is <math>-16</math>.
+Solution: Since we want the <math>a</math>'s where there is only one solution for <math>x</math>, the discriminant has to be <math>0</math>. <math>(a+8)^2-4(4)(9)=a^2+16a-80=0</math>. The sum of these values of <math>a</math> is <math>-16</math>.
 === Intermediate ===

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