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]] [[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]].
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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 | double root]].
  
  

Revision as of 11:24, 10 July 2006

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

  • (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.


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