Difference between revisions of "Quadratic equation"
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Example: Solve the equation <math>x^2-3x+2=0</math> for x. | Example: Solve the equation <math>x^2-3x+2=0</math> for x. | ||
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Solution: <math>x^2-3x+2=0</math> | Solution: <math>x^2-3x+2=0</math> | ||
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First we expand the middle term. This is different for all quadratics. We cleverly choose this so that it has common factors. We now have <math>x^2-x-2x+2=0</math>. | First we expand the middle term. This is different for all quadratics. We cleverly choose this so that it has common factors. We now have <math>x^2-x-2x+2=0</math>. | ||
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Next, we factor out our common terms to get: <math>x(x-1)-2(x-1)=0</math>. | Next, we factor out our common terms to get: <math>x(x-1)-2(x-1)=0</math>. | ||
| − | We can now factor the (x-1) term to get: <math>(x-1)(x-2)=0</math>. By a well known theorem, either <math> (x-1) </math> or <math> (x-2) </math> equals zero. We now have the pair of equations x-1=0, or x-2=0. These give us answers of x=1 or x=2. Plugging these back into the original equation, we find that both of these work! We are done. | + | We can now factor the (x-1) term to get: <math>(x-1)(x-2)=0</math>. By a well known theorem, either <math> (x-1) </math> or <math> (x-2) </math> equals zero. |
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| + | We now have the pair of equations x-1=0, or x-2=0. These give us answers of x=1 or x=2. Plugging these back into the original equation, we find that both of these work! We are done. | ||
=== Completing the square === | === Completing the square === | ||
Revision as of 13:23, 27 June 2006
Quadratic Equations
A quadratic equation is an equation of form 
. a, b, and c are constants, and x is the unknown variable. Quadratic Equations are solved using 3 main strategies: factoring, completing the square, and the quadratic formula.
Factoring
The purpose of factoring is to turn a general quadratic into a product of binomials. This is easier to illustrate than to describe.
Example: Solve the equation 
for x.
Solution:
First we expand the middle term. This is different for all quadratics. We cleverly choose this so that it has common factors. We now have 
.
Next, we factor out our common terms to get: 
.
We can now factor the (x-1) term to get: 
. By a well known theorem, either 
or 
equals zero.
We now have the pair of equations x-1=0, or x-2=0. These give us answers of x=1 or x=2. Plugging these back into the original equation, we find that both of these work! We are done.
Completing the square
Quadratic Formula
See Quadratic Formula.
