Solve Using the Square Root Property (6x+10)^2=125

${(6 x + 10)}^{2} = 125$

Step 1

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

$6 x + 10 = \pm \sqrt{125}$

Step 2

Simplify $\pm \sqrt{125}$ .

Rewrite $125$ as $5^{2} \cdot 5$ .

Factor $25$ out of $125$ .

$6 x + 10 = \pm \sqrt{25 (5)}$

Rewrite $25$ as $5^{2}$ .

$6 x + 10 = \pm \sqrt{5^{2} \cdot 5}$

Pull terms out from under the radical.

$6 x + 10 = \pm 5 \sqrt{5}$

Step 3

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.

$6 x + 10 = 5 \sqrt{5}$

Subtract $10$ from both sides of the equation.

$6 x = 5 \sqrt{5} - 10$

Divide each term in $6 x = 5 \sqrt{5} - 10$ by $6$ and simplify.

Divide each term in $6 x = 5 \sqrt{5} - 10$ by $6$ .

$\frac{6 x}{6} = \frac{5 \sqrt{5}}{6} + \frac{- 10}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 x}{6} = \frac{5 \sqrt{5}}{6} + \frac{- 10}{6}$

Divide $x$ by $1$ .

$x = \frac{5 \sqrt{5}}{6} + \frac{- 10}{6}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 10$ and $6$ .

Factor $2$ out of $- 10$ .

$x = \frac{5 \sqrt{5}}{6} + \frac{2 (- 5)}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

$x = \frac{5 \sqrt{5}}{6} + \frac{2 \cdot - 5}{2 \cdot 3}$

Cancel the common factor.

$x = \frac{5 \sqrt{5}}{6} + \frac{2 \cdot - 5}{2 \cdot 3}$

Rewrite the expression.

$x = \frac{5 \sqrt{5}}{6} + \frac{- 5}{3}$

Move the negative in front of the fraction.

$x = \frac{5 \sqrt{5}}{6} - \frac{5}{3}$

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

$6 x + 10 = - 5 \sqrt{5}$

Subtract $10$ from both sides of the equation.

$6 x = - 5 \sqrt{5} - 10$

Divide each term in $6 x = - 5 \sqrt{5} - 10$ by $6$ and simplify.

Divide each term in $6 x = - 5 \sqrt{5} - 10$ by $6$ .

$\frac{6 x}{6} = \frac{- 5 \sqrt{5}}{6} + \frac{- 10}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 x}{6} = \frac{- 5 \sqrt{5}}{6} + \frac{- 10}{6}$

Divide $x$ by $1$ .

$x = \frac{- 5 \sqrt{5}}{6} + \frac{- 10}{6}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$x = - \frac{5 \sqrt{5}}{6} + \frac{- 10}{6}$

Cancel the common factor of $- 10$ and $6$ .

Factor $2$ out of $- 10$ .

$x = - \frac{5 \sqrt{5}}{6} + \frac{2 (- 5)}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

$x = - \frac{5 \sqrt{5}}{6} + \frac{2 \cdot - 5}{2 \cdot 3}$

Cancel the common factor.

$x = - \frac{5 \sqrt{5}}{6} + \frac{2 \cdot - 5}{2 \cdot 3}$

Rewrite the expression.

$x = - \frac{5 \sqrt{5}}{6} + \frac{- 5}{3}$

Move the negative in front of the fraction.

$x = - \frac{5 \sqrt{5}}{6} - \frac{5}{3}$

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

$x = \frac{5 \sqrt{5}}{6} - \frac{5}{3}, - \frac{5 \sqrt{5}}{6} - \frac{5}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5 \sqrt{5}}{6} - \frac{5}{3}, - \frac{5 \sqrt{5}}{6} - \frac{5}{3}$

Decimal Form:

$x = 0.19672331 \dots, - 3.53005664 \dots$

Solve Using the Square Root Property (6x+10)^2=125

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