Solve Using the Square Root Property (5y+23)^2=32

${(5 y + 23)}^{2} = 32$

Step 1

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

$5 y + 23 = \pm \sqrt{32}$

Step 2

Simplify $\pm \sqrt{32}$ .

Rewrite $32$ as $4^{2} \cdot 2$ .

Factor $16$ out of $32$ .

$5 y + 23 = \pm \sqrt{16 (2)}$

Rewrite $16$ as $4^{2}$ .

$5 y + 23 = \pm \sqrt{4^{2} \cdot 2}$

Pull terms out from under the radical.

$5 y + 23 = \pm 4 \sqrt{2}$

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.

$5 y + 23 = 4 \sqrt{2}$

Subtract $23$ from both sides of the equation.

$5 y = 4 \sqrt{2} - 23$

Divide each term in $5 y = 4 \sqrt{2} - 23$ by $5$ and simplify.

Divide each term in $5 y = 4 \sqrt{2} - 23$ by $5$ .

$\frac{5 y}{5} = \frac{4 \sqrt{2}}{5} + \frac{- 23}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 y}{5} = \frac{4 \sqrt{2}}{5} + \frac{- 23}{5}$

Divide $y$ by $1$ .

$y = \frac{4 \sqrt{2}}{5} + \frac{- 23}{5}$

Simplify the right side.

Move the negative in front of the fraction.

$y = \frac{4 \sqrt{2}}{5} - \frac{23}{5}$

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

$5 y + 23 = - 4 \sqrt{2}$

Subtract $23$ from both sides of the equation.

$5 y = - 4 \sqrt{2} - 23$

Divide each term in $5 y = - 4 \sqrt{2} - 23$ by $5$ and simplify.

Divide each term in $5 y = - 4 \sqrt{2} - 23$ by $5$ .

$\frac{5 y}{5} = \frac{- 4 \sqrt{2}}{5} + \frac{- 23}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 y}{5} = \frac{- 4 \sqrt{2}}{5} + \frac{- 23}{5}$

Divide $y$ by $1$ .

$y = \frac{- 4 \sqrt{2}}{5} + \frac{- 23}{5}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$y = - \frac{4 \sqrt{2}}{5} + \frac{- 23}{5}$

Move the negative in front of the fraction.

$y = - \frac{4 \sqrt{2}}{5} - \frac{23}{5}$

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

$y = \frac{4 \sqrt{2}}{5} - \frac{23}{5}, - \frac{4 \sqrt{2}}{5} - \frac{23}{5}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$y = \frac{4 \sqrt{2}}{5} - \frac{23}{5}, - \frac{4 \sqrt{2}}{5} - \frac{23}{5}$

Decimal Form:

$y = - 3.46862915 \dots, - 5.73137084 \dots$

Solve Using the Square Root Property (5y+23)^2=32

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