Solve Using the Square Root Property (6x+11)^2=125

${(6 x + 11)}^{2} = 125$

Step 1

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

$6 x + 11 = \pm \sqrt{125}$

Step 2

Simplify $\pm \sqrt{125}$ .

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

Factor $25$ out of $125$ .

$6 x + 11 = \pm \sqrt{25 (5)}$

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

$6 x + 11 = \pm \sqrt{5^{2} \cdot 5}$

Pull terms out from under the radical.

$6 x + 11 = \pm 5 \sqrt{5}$

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.

$6 x + 11 = 5 \sqrt{5}$

Subtract $11$ from both sides of the equation.

$6 x = 5 \sqrt{5} - 11$

Divide each term in $6 x = 5 \sqrt{5} - 11$ by $6$ and simplify.

Divide each term in $6 x = 5 \sqrt{5} - 11$ by $6$ .

$\frac{6 x}{6} = \frac{5 \sqrt{5}}{6} + \frac{- 11}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 x}{6} = \frac{5 \sqrt{5}}{6} + \frac{- 11}{6}$

Divide $x$ by $1$ .

$x = \frac{5 \sqrt{5}}{6} + \frac{- 11}{6}$

Simplify the right side.

Move the negative in front of the fraction.

$x = \frac{5 \sqrt{5}}{6} - \frac{11}{6}$

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

$6 x + 11 = - 5 \sqrt{5}$

Subtract $11$ from both sides of the equation.

$6 x = - 5 \sqrt{5} - 11$

Divide each term in $6 x = - 5 \sqrt{5} - 11$ by $6$ and simplify.

Divide each term in $6 x = - 5 \sqrt{5} - 11$ by $6$ .

$\frac{6 x}{6} = \frac{- 5 \sqrt{5}}{6} + \frac{- 11}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 x}{6} = \frac{- 5 \sqrt{5}}{6} + \frac{- 11}{6}$

Divide $x$ by $1$ .

$x = \frac{- 5 \sqrt{5}}{6} + \frac{- 11}{6}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$x = - \frac{5 \sqrt{5}}{6} + \frac{- 11}{6}$

Move the negative in front of the fraction.

$x = - \frac{5 \sqrt{5}}{6} - \frac{11}{6}$

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

$x = \frac{5 \sqrt{5}}{6} - \frac{11}{6}, - \frac{5 \sqrt{5}}{6} - \frac{11}{6}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5 \sqrt{5}}{6} - \frac{11}{6}, - \frac{5 \sqrt{5}}{6} - \frac{11}{6}$

Decimal Form:

$x = 0.03005664 \dots, - 3.69672331 \dots$

Solve Using the Square Root Property (6x+11)^2=125

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