Solve Using the Square Root Property (6y+15)^2=125

${(6 y + 15)}^{2} = 125$

Step 1

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

$6 y + 15 = \pm \sqrt{125}$

Step 2

Simplify $\pm \sqrt{125}$ .

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

Factor $25$ out of $125$ .

$6 y + 15 = \pm \sqrt{25 (5)}$

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

$6 y + 15 = \pm \sqrt{5^{2} \cdot 5}$

Pull terms out from under the radical.

$6 y + 15 = \pm 5 \sqrt{5}$

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.

$6 y + 15 = 5 \sqrt{5}$

Subtract $15$ from both sides of the equation.

$6 y = 5 \sqrt{5} - 15$

Divide each term in $6 y = 5 \sqrt{5} - 15$ by $6$ and simplify.

Divide each term in $6 y = 5 \sqrt{5} - 15$ by $6$ .

$\frac{6 y}{6} = \frac{5 \sqrt{5}}{6} + \frac{- 15}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 y}{6} = \frac{5 \sqrt{5}}{6} + \frac{- 15}{6}$

Divide $y$ by $1$ .

$y = \frac{5 \sqrt{5}}{6} + \frac{- 15}{6}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 15$ and $6$ .

Factor $3$ out of $- 15$ .

$y = \frac{5 \sqrt{5}}{6} + \frac{3 (- 5)}{6}$

Cancel the common factors.

Factor $3$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

$y = \frac{5 \sqrt{5}}{6} + \frac{- 5}{2}$

Move the negative in front of the fraction.

$y = \frac{5 \sqrt{5}}{6} - \frac{5}{2}$

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

$6 y + 15 = - 5 \sqrt{5}$

Subtract $15$ from both sides of the equation.

$6 y = - 5 \sqrt{5} - 15$

Divide each term in $6 y = - 5 \sqrt{5} - 15$ by $6$ and simplify.

Divide each term in $6 y = - 5 \sqrt{5} - 15$ by $6$ .

$\frac{6 y}{6} = \frac{- 5 \sqrt{5}}{6} + \frac{- 15}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 y}{6} = \frac{- 5 \sqrt{5}}{6} + \frac{- 15}{6}$

Divide $y$ by $1$ .

$y = \frac{- 5 \sqrt{5}}{6} + \frac{- 15}{6}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$y = - \frac{5 \sqrt{5}}{6} + \frac{- 15}{6}$

Cancel the common factor of $- 15$ and $6$ .

Factor $3$ out of $- 15$ .

$y = - \frac{5 \sqrt{5}}{6} + \frac{3 (- 5)}{6}$

Cancel the common factors.

Factor $3$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

$y = - \frac{5 \sqrt{5}}{6} + \frac{- 5}{2}$

Move the negative in front of the fraction.

$y = - \frac{5 \sqrt{5}}{6} - \frac{5}{2}$

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

$y = \frac{5 \sqrt{5}}{6} - \frac{5}{2}, - \frac{5 \sqrt{5}}{6} - \frac{5}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$y = \frac{5 \sqrt{5}}{6} - \frac{5}{2}, - \frac{5 \sqrt{5}}{6} - \frac{5}{2}$

Decimal Form:

$y = - 0.63661001 \dots, - 4.36338998 \dots$

Solve Using the Square Root Property (6y+15)^2=125

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