Solve Using the Square Root Property (2x+1)^2=12

${(2 x + 1)}^{2} = 12$

Step 1

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

$2 x + 1 = \pm \sqrt{12}$

Step 2

Simplify $\pm \sqrt{12}$ .

Rewrite $12$ as $2^{2} \cdot 3$ .

Factor $4$ out of $12$ .

$2 x + 1 = \pm \sqrt{4 (3)}$

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

$2 x + 1 = \pm \sqrt{2^{2} \cdot 3}$

Pull terms out from under the radical.

$2 x + 1 = \pm 2 \sqrt{3}$

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.

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

Subtract $1$ from both sides of the equation.

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

Divide each term in $2 x = 2 \sqrt{3} - 1$ by $2$ and simplify.

Divide each term in $2 x = 2 \sqrt{3} - 1$ by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Simplify each term.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $\sqrt{3}$ by $1$ .

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

Move the negative in front of the fraction.

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

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

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

Subtract $1$ from both sides of the equation.

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

Divide each term in $2 x = - 2 \sqrt{3} - 1$ by $2$ and simplify.

Divide each term in $2 x = - 2 \sqrt{3} - 1$ by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Simplify each term.

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

Factor $2$ out of $- 2 \sqrt{3}$ .

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $- \sqrt{3}$ by $1$ .

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

Move the negative in front of the fraction.

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

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

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 1.23205080 \dots, - 2.23205080 \dots$

Solve Using the Square Root Property (2x+1)^2=12

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