Solve Using the Square Root Property (5m-6)^2=7

${(5 m - 6)}^{2} = 7$

Step 1

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

$5 m - 6 = \pm \sqrt{7}$

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.

$5 m - 6 = \sqrt{7}$

Add $6$ to both sides of the equation.

$5 m = \sqrt{7} + 6$

Divide each term in $5 m = \sqrt{7} + 6$ by $5$ and simplify.

Divide each term in $5 m = \sqrt{7} + 6$ by $5$ .

$\frac{5 m}{5} = \frac{\sqrt{7}}{5} + \frac{6}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 m}{5} = \frac{\sqrt{7}}{5} + \frac{6}{5}$

Divide $m$ by $1$ .

$m = \frac{\sqrt{7}}{5} + \frac{6}{5}$

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

$5 m - 6 = - \sqrt{7}$

Add $6$ to both sides of the equation.

$5 m = - \sqrt{7} + 6$

Divide each term in $5 m = - \sqrt{7} + 6$ by $5$ and simplify.

Divide each term in $5 m = - \sqrt{7} + 6$ by $5$ .

$\frac{5 m}{5} = \frac{- \sqrt{7}}{5} + \frac{6}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 m}{5} = \frac{- \sqrt{7}}{5} + \frac{6}{5}$

Divide $m$ by $1$ .

$m = \frac{- \sqrt{7}}{5} + \frac{6}{5}$

Simplify the right side.

Move the negative in front of the fraction.

$m = - \frac{\sqrt{7}}{5} + \frac{6}{5}$

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

$m = \frac{\sqrt{7}}{5} + \frac{6}{5}, - \frac{\sqrt{7}}{5} + \frac{6}{5}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$m = \frac{\sqrt{7}}{5} + \frac{6}{5}, - \frac{\sqrt{7}}{5} + \frac{6}{5}$

Decimal Form:

$m = 1.72915026 \dots, 0.67084973 \dots$