Difference between revisions of "Discriminant"
m (fixed LaTeX) |
Pianoforte (talk | contribs) (example) |
||
| Line 1: | Line 1: | ||
The '''discriminant''' of a [[Quadratic Equations | Quadratic Equation]] of the form <math>a{x}^2+b{x}+{c}=0</math> is the quantity <math>b^2-4ac</math>. When <math>{a},{b},{c}</math> are real, this is a notable quantity, because if the discriminant is positive, the equation has two [[real]] [[Roots | 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 | double root]]. | The '''discriminant''' of a [[Quadratic Equations | Quadratic Equation]] of the form <math>a{x}^2+b{x}+{c}=0</math> is the quantity <math>b^2-4ac</math>. When <math>{a},{b},{c}</math> are real, this is a notable quantity, because if the discriminant is positive, the equation has two [[real]] [[Roots | 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 | double root]]. | ||
| + | |||
| + | |||
| + | ==Example== | ||
| + | * (AMC 12 2005) There are two values of a for which the equation <math>4x^2+ax+8x+9=0</math> 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. <math>(a+8)^2-4\times4\times9=a^2+16a-80=0</math>. The sum of these values of a is -16. | ||
== Other resources == | == Other resources == | ||
* [http://en.wikipedia.org/wiki/Discriminant Wikipedia entry] | * [http://en.wikipedia.org/wiki/Discriminant Wikipedia entry] | ||
Revision as of 10:32, 18 June 2006
The discriminant of a Quadratic Equation of the form 
is the quantity 
. When 
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
- (AMC 12 2005) There are two values of a for which the equation
has only one solution for x. What is the sum of these values of a?
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. 
. The sum of these values of a is -16.
