Equations with Squared Variables

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

1. If \(n\) is even natural number and \(x^n = a^n\), then \(|x|=|a|\); \(a\) is Real number.

2. If \(n\) is odd natural number and \(x^n = a^n\), then \(x=a\); \(a\) is Real number.

3. 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 rule 2

The roots are \(x=1\) or \(x=-1\).