Difference between revisions of "Quadratic equation"
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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 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= | + | 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. |
=== Completing the square === | === Completing the square === | ||
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* [[Factoring Quadratics]] | * [[Factoring Quadratics]] | ||
+ | [[Category:Algebra]] | ||
+ | [[Category:Quadratic equations]] | ||
[[Category:Definition]] | [[Category:Definition]] | ||
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Latest revision as of 12:04, 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.