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=== Quadratic Equations ===
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]].
 
A quadratic equation is an equation of form <math> {a}{x}^2+{b}{x}+{c}=0</math>. 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 ===
=== Factoring ===
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 binomials. This is easier to illustrate than to describe.
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.


Example: Solve the equation <math>x^2-3x+2=0</math> for x.
Solution: <math>x^2-3x+2=0</math>


Solution: <math>x^2-3x+2=0</math>
First, we expand the middle term: <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>.
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 the zero-product property, either <math> (x-1) </math> or <math> (x-2) </math> equals zero.  
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 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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=== Quadratic Formula ===
=== Quadratic Formula ===
See [[Quadratic Formula]].
See [[Quadratic Formula]].
== See Also ==
* [[Discriminant]]
* [[Vieta's Formulas]]
* [[Quadratic Inequality]]
* [[Factoring Quadratics]]
[[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 ${a}{x}^2+{b}{x}+{c}=0$, where $a$, $b$ and $c$ are constants (that is, they do not depend on $x$) and $x$ 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 $x^2-3x+2=0$ for $x$. Note: This is different for all quadratics; we cleverly chose this so that it has common factors.

Solution: $x^2-3x+2=0$

First, we expand the middle term: $x^2-x-2x+2=0$.

Next, we factor out our common terms to get $x(x-1)-2(x-1)=0$.

We can now factor the $(x-1)$ term to get $(x-1)(x-2)=0$. By the zero-product property, either $(x-1)$ or $(x-2)$ equals zero.

We now have the pair of equations $x-1=0$ and $x-2=0$. These give us the answers $x=1$ and $x=2$, which can also be written as $x=\{1,\,2\}$. Plugging these back into the original equation, we find that both of these work! We are done.

Completing the square

Completing the square

Quadratic Formula

See Quadratic Formula.

See Also

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