Difference between revisions of "Quadratic equation"
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| − | A '''quadratic equation''' is an [[equation]] of the form <math> {a}{x}^2+{b}{x}+{c}=0</math>, where <math>a</math>, <math>b</math> and <math>c</math> are [[constant]]s and <math>x</math> is the unknown | + | A '''quadratic equation''' in one [[variable]] is an [[equation]] of the form <math> {a}{x}^2+{b}{x}+{c}=0</math>, where <math>a</math>, <math>b</math> and <math>c</math> are [[constant]]s (that is, they do not depend on <math>x</math>) and <math>x</math> is the unknown variable. Quadratic equations are solved using one of three main strategies: [[factoring]], [[completing the square]] and the [[quadratic formula]]. |
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=== Factoring === | === Factoring === | ||
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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. | ||
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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 <math>(x-1)</math> term to get <math>(x-1)(x-2)=0</math>. By | + | We can now factor the <math>(x-1)</math> term to get <math>(x-1)(x-2)=0</math>. By the zero-product property, either <math> (x-1) </math> or <math> (x-2) </math> equals zero. |
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. | 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. | ||
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=== Completing the square === | === Completing the square === | ||
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== See Also == | == See Also == | ||
* [[Discriminant]] | * [[Discriminant]] | ||
| + | * [[Vieta's Formulas]] | ||
* [[Quadratic Inequality]] | * [[Quadratic Inequality]] | ||
| + | * [[Factoring Quadratics]] | ||
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| + | [[Category:Algebra]] | ||
| + | [[Category:Quadratic equations]] | ||
| + | [[Category:Definition]] | ||
Latest revision as of 11:04, 15 July 2021
A quadratic equation in one variable is an equation of the form 
, where 
, 
and 
are constants (that is, they do not depend on 
) and is the unknown variable. Quadratic equations are solved using one of three 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 the zero-product property, 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.
