Solve Using the Square Root Property (2x+3)^2=27

${(2 x + 3)}^{2} = 27$

Step 1

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

$2 x + 3 = \pm \sqrt{27}$

Step 2

Simplify $\pm \sqrt{27}$ .

Rewrite $27$ as $3^{2} \cdot 3$ .

Factor $9$ out of $27$ .

$2 x + 3 = \pm \sqrt{9 (3)}$

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

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

Pull terms out from under the radical.

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

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.

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

Subtract $3$ from both sides of the equation.

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

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

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

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

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

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

Subtract $3$ from both sides of the equation.

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

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

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

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

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

Move the negative in front of the fraction.

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

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

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 1.09807621 \dots, - 4.09807621 \dots$

Solve Using the Square Root Property (2x+3)^2=27

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