# Equations with Squared Variables

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To solved equation with *Squared Variables*, we need to used the basic operation in Algebra and this is *nth root both side*.

Basic *nth root both side rule*

If $n$ is even natural number and $x^n = a^n$, then $|x|=|a|$; $a$ is

*Real number*.If $n$ is odd natural number and $x^n = a^n$, then $x=a$; $a$ is

*Real number*.If $n$ is natural number and $x^n = a^n$, then $x = a$; $a \geq 0$

Another tool to solve equation involving *Squared Variables* is the basic property of absolute value.

$|x|=a => x = a$ or $x = -a$.

## Find the value of $x$ if $x<0$ and $x^2 - 9=0$.

$x^2-9=0 => x^2 = 9 => x^2 = 3^2$

By Basic

nth root both side rule#1$=> |x| = |3| => x = 3$ or $x = -3$

Since, $x<0 => x = -3$

## Find the value of x if $x^2+3=13$

$x^2+3=13 => x^2 = 10 => x^2 = (\sqrt{10})^2$

By Basic

nth root both side rule#1$=> |x|= | \sqrt{10}|$

Therefore, $x= \sqrt{10}$ or $x= - \sqrt{10}$

## Find $x$ if x^6 = 1)

$x^6=1=> (x^3)^2 = (1^3)^2 => |x^3| = |1^3| => x^3=(1)^3$ or $x^3 =-(1)^3$

By Basic

nth root both side rule2The roots are $x=1$ or $x=-1$.

**Cite as:**Equations with Squared Variables.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/solving-equations-squared-variables/