Solve Using the Square Root Property (13x-7)^2=110

${(13 x - 7)}^{2} = 110$

Step 1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$13 x - 7 = \pm \sqrt{110}$

Step 2

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$13 x - 7 = \sqrt{110}$

Add $7$ to both sides of the equation.

$13 x = \sqrt{110} + 7$

Divide each term in $13 x = \sqrt{110} + 7$ by $13$ and simplify.

Divide each term in $13 x = \sqrt{110} + 7$ by $13$ .

$\frac{13 x}{13} = \frac{\sqrt{110}}{13} + \frac{7}{13}$

Simplify the left side.

Cancel the common factor of $13$ .

Cancel the common factor.

$\frac{13 x}{13} = \frac{\sqrt{110}}{13} + \frac{7}{13}$

Divide $x$ by $1$ .

$x = \frac{\sqrt{110}}{13} + \frac{7}{13}$

Next, use the negative value of the $\pm$ to find the second solution.

$13 x - 7 = - \sqrt{110}$

Add $7$ to both sides of the equation.

$13 x = - \sqrt{110} + 7$

Divide each term in $13 x = - \sqrt{110} + 7$ by $13$ and simplify.

Divide each term in $13 x = - \sqrt{110} + 7$ by $13$ .

$\frac{13 x}{13} = \frac{- \sqrt{110}}{13} + \frac{7}{13}$

Simplify the left side.

Cancel the common factor of $13$ .

Cancel the common factor.

$\frac{13 x}{13} = \frac{- \sqrt{110}}{13} + \frac{7}{13}$

Divide $x$ by $1$ .

$x = \frac{- \sqrt{110}}{13} + \frac{7}{13}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{\sqrt{110}}{13} + \frac{7}{13}$

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{\sqrt{110}}{13} + \frac{7}{13}, - \frac{\sqrt{110}}{13} + \frac{7}{13}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\sqrt{110}}{13} + \frac{7}{13}, - \frac{\sqrt{110}}{13} + \frac{7}{13}$

Decimal Form:

$x = 1.34523757 \dots, - 0.26831449 \dots$