Solve Using the Square Root Property (3x+1)^2=27

${(3 x + 1)}^{2} = 27$

Step 1

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

$3 x + 1 = \pm \sqrt{27}$

Step 2

Simplify $\pm \sqrt{27}$ .

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

Factor $9$ out of $27$ .

$3 x + 1 = \pm \sqrt{9 (3)}$

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

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

Pull terms out from under the radical.

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

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.

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

Subtract $1$ from both sides of the equation.

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

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

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

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Simplify each term.

Cancel the common factor of $3$ .

Cancel the common factor.

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

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

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

Move the negative in front of the fraction.

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

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

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

Subtract $1$ from both sides of the equation.

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

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

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

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 3$ and $3$ .

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

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

Cancel the common factors.

Factor $3$ out of $3$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Move the negative in front of the fraction.

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

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

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 1.39871747 \dots, - 2.06538414 \dots$

Solve Using the Square Root Property (3x+1)^2=27

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