Solve Using the Square Root Property (8y+22)^2=75

${(8 y + 22)}^{2} = 75$

Step 1

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

$8 y + 22 = \pm \sqrt{75}$

Step 2

Simplify $\pm \sqrt{75}$ .

Rewrite $75$ as $5^{2} \cdot 3$ .

Factor $25$ out of $75$ .

$8 y + 22 = \pm \sqrt{25 (3)}$

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

$8 y + 22 = \pm \sqrt{5^{2} \cdot 3}$

Pull terms out from under the radical.

$8 y + 22 = \pm 5 \sqrt{3}$

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.

$8 y + 22 = 5 \sqrt{3}$

Subtract $22$ from both sides of the equation.

$8 y = 5 \sqrt{3} - 22$

Divide each term in $8 y = 5 \sqrt{3} - 22$ by $8$ and simplify.

Divide each term in $8 y = 5 \sqrt{3} - 22$ by $8$ .

$\frac{8 y}{8} = \frac{5 \sqrt{3}}{8} + \frac{- 22}{8}$

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 y}{8} = \frac{5 \sqrt{3}}{8} + \frac{- 22}{8}$

Divide $y$ by $1$ .

$y = \frac{5 \sqrt{3}}{8} + \frac{- 22}{8}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 22$ and $8$ .

Factor $2$ out of $- 22$ .

$y = \frac{5 \sqrt{3}}{8} + \frac{2 (- 11)}{8}$

Cancel the common factors.

Factor $2$ out of $8$ .

$y = \frac{5 \sqrt{3}}{8} + \frac{2 \cdot - 11}{2 \cdot 4}$

Cancel the common factor.

$y = \frac{5 \sqrt{3}}{8} + \frac{2 \cdot - 11}{2 \cdot 4}$

Rewrite the expression.

$y = \frac{5 \sqrt{3}}{8} + \frac{- 11}{4}$

Move the negative in front of the fraction.

$y = \frac{5 \sqrt{3}}{8} - \frac{11}{4}$

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

$8 y + 22 = - 5 \sqrt{3}$

Subtract $22$ from both sides of the equation.

$8 y = - 5 \sqrt{3} - 22$

Divide each term in $8 y = - 5 \sqrt{3} - 22$ by $8$ and simplify.

Divide each term in $8 y = - 5 \sqrt{3} - 22$ by $8$ .

$\frac{8 y}{8} = \frac{- 5 \sqrt{3}}{8} + \frac{- 22}{8}$

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 y}{8} = \frac{- 5 \sqrt{3}}{8} + \frac{- 22}{8}$

Divide $y$ by $1$ .

$y = \frac{- 5 \sqrt{3}}{8} + \frac{- 22}{8}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$y = - \frac{5 \sqrt{3}}{8} + \frac{- 22}{8}$

Cancel the common factor of $- 22$ and $8$ .

Factor $2$ out of $- 22$ .

$y = - \frac{5 \sqrt{3}}{8} + \frac{2 (- 11)}{8}$

Cancel the common factors.

Factor $2$ out of $8$ .

$y = - \frac{5 \sqrt{3}}{8} + \frac{2 \cdot - 11}{2 \cdot 4}$

Cancel the common factor.

$y = - \frac{5 \sqrt{3}}{8} + \frac{2 \cdot - 11}{2 \cdot 4}$

Rewrite the expression.

$y = - \frac{5 \sqrt{3}}{8} + \frac{- 11}{4}$

Move the negative in front of the fraction.

$y = - \frac{5 \sqrt{3}}{8} - \frac{11}{4}$

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

$y = \frac{5 \sqrt{3}}{8} - \frac{11}{4}, - \frac{5 \sqrt{3}}{8} - \frac{11}{4}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$y = \frac{5 \sqrt{3}}{8} - \frac{11}{4}, - \frac{5 \sqrt{3}}{8} - \frac{11}{4}$

Decimal Form:

$y = - 1.66746824 \dots, - 3.83253175 \dots$

Solve Using the Square Root Property (8y+22)^2=75

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