# 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/