Solve Using the Square Root Property (2x+3)^2=32

${(2 x + 3)}^{2} = 32$

Step 1

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

$2 x + 3 = \pm \sqrt{32}$

Step 2

Simplify $\pm \sqrt{32}$ .

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

Factor $16$ out of $32$ .

$2 x + 3 = \pm \sqrt{16 (2)}$

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

$2 x + 3 = \pm \sqrt{4^{2} \cdot 2}$

Pull terms out from under the radical.

$2 x + 3 = \pm 4 \sqrt{2}$

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.

$2 x + 3 = 4 \sqrt{2}$

Subtract $3$ from both sides of the equation.

$2 x = 4 \sqrt{2} - 3$

Divide each term in $2 x = 4 \sqrt{2} - 3$ by $2$ and simplify.

Divide each term in $2 x = 4 \sqrt{2} - 3$ by $2$ .

$\frac{2 x}{2} = \frac{4 \sqrt{2}}{2} + \frac{- 3}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{4 \sqrt{2}}{2} + \frac{- 3}{2}$

Divide $x$ by $1$ .

$x = \frac{4 \sqrt{2}}{2} + \frac{- 3}{2}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $4$ and $2$ .

Factor $2$ out of $4 \sqrt{2}$ .

$x = \frac{2 (2 \sqrt{2})}{2} + \frac{- 3}{2}$

Cancel the common factors.

Factor $2$ out of $2$ .

$x = \frac{2 (2 \sqrt{2})}{2 (1)} + \frac{- 3}{2}$

Cancel the common factor.

$x = \frac{2 (2 \sqrt{2})}{2 \cdot 1} + \frac{- 3}{2}$

Rewrite the expression.

$x = \frac{2 \sqrt{2}}{1} + \frac{- 3}{2}$

Divide $2 \sqrt{2}$ by $1$ .

$x = 2 \sqrt{2} + \frac{- 3}{2}$

Move the negative in front of the fraction.

$x = 2 \sqrt{2} - \frac{3}{2}$

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

$2 x + 3 = - 4 \sqrt{2}$

Subtract $3$ from both sides of the equation.

$2 x = - 4 \sqrt{2} - 3$

Divide each term in $2 x = - 4 \sqrt{2} - 3$ by $2$ and simplify.

Divide each term in $2 x = - 4 \sqrt{2} - 3$ by $2$ .

$\frac{2 x}{2} = \frac{- 4 \sqrt{2}}{2} + \frac{- 3}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 4 \sqrt{2}}{2} + \frac{- 3}{2}$

Divide $x$ by $1$ .

$x = \frac{- 4 \sqrt{2}}{2} + \frac{- 3}{2}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 4$ and $2$ .

Factor $2$ out of $- 4 \sqrt{2}$ .

$x = \frac{2 (- 2 \sqrt{2})}{2} + \frac{- 3}{2}$

Cancel the common factors.

Factor $2$ out of $2$ .

$x = \frac{2 (- 2 \sqrt{2})}{2 (1)} + \frac{- 3}{2}$

Cancel the common factor.

$x = \frac{2 (- 2 \sqrt{2})}{2 \cdot 1} + \frac{- 3}{2}$

Rewrite the expression.

$x = \frac{- 2 \sqrt{2}}{1} + \frac{- 3}{2}$

Divide $- 2 \sqrt{2}$ by $1$ .

$x = - 2 \sqrt{2} + \frac{- 3}{2}$

Move the negative in front of the fraction.

$x = - 2 \sqrt{2} - \frac{3}{2}$

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

$x = 2 \sqrt{2} - \frac{3}{2}, - 2 \sqrt{2} - \frac{3}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = 2 \sqrt{2} - \frac{3}{2}, - 2 \sqrt{2} - \frac{3}{2}$

Decimal Form:

$x = 1.32842712 \dots, - 4.32842712 \dots$

Solve Using the Square Root Property (2x+3)^2=32

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