Solve Using the Square Root Property (8y+14)^2=45

${(8 y + 14)}^{2} = 45$

Step 1

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

$8 y + 14 = \pm \sqrt{45}$

Step 2

Simplify $\pm \sqrt{45}$ .

Rewrite $45$ as $3^{2} \cdot 5$ .

Factor $9$ out of $45$ .

$8 y + 14 = \pm \sqrt{9 (5)}$

Rewrite $9$ as $3^{2}$ .

$8 y + 14 = \pm \sqrt{3^{2} \cdot 5}$

Pull terms out from under the radical.

$8 y + 14 = \pm 3 \sqrt{5}$

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.

$8 y + 14 = 3 \sqrt{5}$

Subtract $14$ from both sides of the equation.

$8 y = 3 \sqrt{5} - 14$

Divide each term in $8 y = 3 \sqrt{5} - 14$ by $8$ and simplify.

Divide each term in $8 y = 3 \sqrt{5} - 14$ by $8$ .

$\frac{8 y}{8} = \frac{3 \sqrt{5}}{8} + \frac{- 14}{8}$

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 y}{8} = \frac{3 \sqrt{5}}{8} + \frac{- 14}{8}$

Divide $y$ by $1$ .

$y = \frac{3 \sqrt{5}}{8} + \frac{- 14}{8}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 14$ and $8$ .

Factor $2$ out of $- 14$ .

$y = \frac{3 \sqrt{5}}{8} + \frac{2 (- 7)}{8}$

Cancel the common factors.

Factor $2$ out of $8$ .

$y = \frac{3 \sqrt{5}}{8} + \frac{2 \cdot - 7}{2 \cdot 4}$

Cancel the common factor.

$y = \frac{3 \sqrt{5}}{8} + \frac{2 \cdot - 7}{2 \cdot 4}$

Rewrite the expression.

$y = \frac{3 \sqrt{5}}{8} + \frac{- 7}{4}$

Move the negative in front of the fraction.

$y = \frac{3 \sqrt{5}}{8} - \frac{7}{4}$

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

$8 y + 14 = - 3 \sqrt{5}$

Subtract $14$ from both sides of the equation.

$8 y = - 3 \sqrt{5} - 14$

Divide each term in $8 y = - 3 \sqrt{5} - 14$ by $8$ and simplify.

Divide each term in $8 y = - 3 \sqrt{5} - 14$ by $8$ .

$\frac{8 y}{8} = \frac{- 3 \sqrt{5}}{8} + \frac{- 14}{8}$

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 y}{8} = \frac{- 3 \sqrt{5}}{8} + \frac{- 14}{8}$

Divide $y$ by $1$ .

$y = \frac{- 3 \sqrt{5}}{8} + \frac{- 14}{8}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$y = - \frac{3 \sqrt{5}}{8} + \frac{- 14}{8}$

Cancel the common factor of $- 14$ and $8$ .

Factor $2$ out of $- 14$ .

$y = - \frac{3 \sqrt{5}}{8} + \frac{2 (- 7)}{8}$

Cancel the common factors.

Factor $2$ out of $8$ .

$y = - \frac{3 \sqrt{5}}{8} + \frac{2 \cdot - 7}{2 \cdot 4}$

Cancel the common factor.

$y = - \frac{3 \sqrt{5}}{8} + \frac{2 \cdot - 7}{2 \cdot 4}$

Rewrite the expression.

$y = - \frac{3 \sqrt{5}}{8} + \frac{- 7}{4}$

Move the negative in front of the fraction.

$y = - \frac{3 \sqrt{5}}{8} - \frac{7}{4}$

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

$y = \frac{3 \sqrt{5}}{8} - \frac{7}{4}, - \frac{3 \sqrt{5}}{8} - \frac{7}{4}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$y = \frac{3 \sqrt{5}}{8} - \frac{7}{4}, - \frac{3 \sqrt{5}}{8} - \frac{7}{4}$

Decimal Form:

$y = - 0.91147450 \dots, - 2.58852549 \dots$

Solve Using the Square Root Property (8y+14)^2=45

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