Difference between revisions of "Quadratic equation"
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| − | A '''quadratic equation''' is an [[equation]] of form <math> {a}{x}^2+{b}{x}+{c}=0</math>. a, b, and c are [[constant]]s, and x is the unknown [[variable]]. Quadratic equations are solved using 3 main strategies: [[factoring]], [[completing the square]], and the [[quadratic formula]]. | + | A '''quadratic equation''' is an [[equation]] of the form <math> {a}{x}^2+{b}{x}+{c}=0</math>. a, b, and c are [[constant]]s, and x is the unknown [[variable]]. Quadratic equations are solved using 3 main strategies: [[factoring]], [[completing the square]], and the [[quadratic formula]]. |
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The purpose of factoring is to turn a general quadratic into a product of [[binomial]]s. This is easier to illustrate than to describe. | The purpose of factoring is to turn a general quadratic into a product of [[binomial]]s. This is easier to illustrate than to describe. | ||
| − | 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 <math>x</math>. Note: This is different for all quadratics; we cleverly chose this so that it has common factors. |
Solution: <math>x^2-3x+2=0</math> | Solution: <math>x^2-3x+2=0</math> | ||
| − | First we expand the middle term | + | First, we expand the middle term: <math>x^2-x-2x+2=0</math>. |
| − | Next, we factor out our common terms to get | + | Next, we factor out our common terms to get <math>x(x-1)-2(x-1)=0</math>. |
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| − | We now have the pair of equations x-1=0 | + | We can now factor the <math>(x-1)</math> 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 <math>x-1=0</math> and <math>x-2=0</math>. These give us the answers <math>x=1</math> and <math>x=2</math>, which can also be written as <math>x=\{1,\,2\}</math>. Plugging these back into the original equation, we find that both of these work! We are done. | ||
Revision as of 02:37, 30 October 2006
A quadratic equation is an equation of the 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 
. Note: This is different for all quadratics; we cleverly chose this so that it has common factors.
Solution:
First, we expand the middle term: 
.
Next, we factor out our common terms to get 
.
We can now factor the 
term to get 
. By a well known theorem, either or 
equals zero.
We now have the pair of equations 
and 
. These give us the answers 
and 
, which can also be written as 
. 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.
