Difference between revisions of "Discriminant"
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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]] [[root]]s; if the discriminant is negative, the equation has two [[nonreal]] roots; and if the discriminant is 0, the equation has a [[real]] [[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]] [[root]]s; if the discriminant is negative, the equation has two [[nonreal]] roots; and if the discriminant is 0, the equation has a [[real]] [[double root]]. | ||
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== Example Problems == | == Example Problems == | ||
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=== Intermediate === | === Intermediate === | ||
* [[1977_Canadian_MO_Problems/Problem_1 | 1977 Canadian MO Problem 1]] | * [[1977_Canadian_MO_Problems/Problem_1 | 1977 Canadian MO Problem 1]] | ||
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== Other resources == | == Other resources == | ||
* [http://en.wikipedia.org/wiki/Discriminant Wikipedia entry] | * [http://en.wikipedia.org/wiki/Discriminant Wikipedia entry] | ||
Revision as of 20:53, 21 April 2008
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 Problems
Introductory
- (AMC 12 2005) There are two values of
for which the equation 
has only one solution for 
. What is the sum of these values of ?
has only one solution for Solution: Since we want the 
's where there is only one solution for 
, the discriminant has to be 
. 
. The sum of these values of is 
.
