Solve Using the Square Root Property (6x+2)^2+4=28

${(6 x + 2)}^{2} + 4 = 28$

Step 1

Subtract $4$ from both sides of the equation.

${(6 x + 2)}^{2} = 28 - 4$

Step 2

Subtract $4$ from $28$ .

${(6 x + 2)}^{2} = 24$

Step 3

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

$6 x + 2 = \pm \sqrt{24}$

Step 4

Simplify $\pm \sqrt{24}$ .

Rewrite $24$ as $2^{2} \cdot 6$ .

Factor $4$ out of $24$ .

$6 x + 2 = \pm \sqrt{4 (6)}$

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

$6 x + 2 = \pm \sqrt{2^{2} \cdot 6}$

Pull terms out from under the radical.

$6 x + 2 = \pm 2 \sqrt{6}$

Step 5

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.

$6 x + 2 = 2 \sqrt{6}$

Subtract $2$ from both sides of the equation.

$6 x = 2 \sqrt{6} - 2$

Divide each term in $6 x = 2 \sqrt{6} - 2$ by $6$ and simplify.

Divide each term in $6 x = 2 \sqrt{6} - 2$ by $6$ .

$\frac{6 x}{6} = \frac{2 \sqrt{6}}{6} + \frac{- 2}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 x}{6} = \frac{2 \sqrt{6}}{6} + \frac{- 2}{6}$

Divide $x$ by $1$ .

$x = \frac{2 \sqrt{6}}{6} + \frac{- 2}{6}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $2$ and $6$ .

Factor $2$ out of $2 \sqrt{6}$ .

$x = \frac{2 \sqrt{6}}{6} + \frac{- 2}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

$x = \frac{2 \sqrt{6}}{2 \cdot 3} + \frac{- 2}{6}$

Cancel the common factor.

$x = \frac{2 \sqrt{6}}{2 \cdot 3} + \frac{- 2}{6}$

Rewrite the expression.

$x = \frac{\sqrt{6}}{3} + \frac{- 2}{6}$

Cancel the common factor of $- 2$ and $6$ .

Factor $2$ out of $- 2$ .

$x = \frac{\sqrt{6}}{3} + \frac{2 (- 1)}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

$x = \frac{\sqrt{6}}{3} + \frac{2 \cdot - 1}{2 \cdot 3}$

Cancel the common factor.

$x = \frac{\sqrt{6}}{3} + \frac{2 \cdot - 1}{2 \cdot 3}$

Rewrite the expression.

$x = \frac{\sqrt{6}}{3} + \frac{- 1}{3}$

Move the negative in front of the fraction.

$x = \frac{\sqrt{6}}{3} - \frac{1}{3}$

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

$6 x + 2 = - 2 \sqrt{6}$

Subtract $2$ from both sides of the equation.

$6 x = - 2 \sqrt{6} - 2$

Divide each term in $6 x = - 2 \sqrt{6} - 2$ by $6$ and simplify.

Divide each term in $6 x = - 2 \sqrt{6} - 2$ by $6$ .

$\frac{6 x}{6} = \frac{- 2 \sqrt{6}}{6} + \frac{- 2}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 x}{6} = \frac{- 2 \sqrt{6}}{6} + \frac{- 2}{6}$

Divide $x$ by $1$ .

$x = \frac{- 2 \sqrt{6}}{6} + \frac{- 2}{6}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 2$ and $6$ .

Factor $2$ out of $- 2 \sqrt{6}$ .

$x = \frac{2 (- \sqrt{6})}{6} + \frac{- 2}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

$x = \frac{2 (- \sqrt{6})}{2 (3)} + \frac{- 2}{6}$

Cancel the common factor.

$x = \frac{2 (- \sqrt{6})}{2 \cdot 3} + \frac{- 2}{6}$

Rewrite the expression.

$x = \frac{- \sqrt{6}}{3} + \frac{- 2}{6}$

Move the negative in front of the fraction.

$x = - \frac{\sqrt{6}}{3} + \frac{- 2}{6}$

Cancel the common factor of $- 2$ and $6$ .

Factor $2$ out of $- 2$ .

$x = - \frac{\sqrt{6}}{3} + \frac{2 (- 1)}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

$x = - \frac{\sqrt{6}}{3} + \frac{2 \cdot - 1}{2 \cdot 3}$

Cancel the common factor.

$x = - \frac{\sqrt{6}}{3} + \frac{2 \cdot - 1}{2 \cdot 3}$

Rewrite the expression.

$x = - \frac{\sqrt{6}}{3} + \frac{- 1}{3}$

Move the negative in front of the fraction.

$x = - \frac{\sqrt{6}}{3} - \frac{1}{3}$

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

$x = \frac{\sqrt{6}}{3} - \frac{1}{3}, - \frac{\sqrt{6}}{3} - \frac{1}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\sqrt{6}}{3} - \frac{1}{3}, - \frac{\sqrt{6}}{3} - \frac{1}{3}$

Decimal Form:

$x = 0.48316324 \dots, - 1.14982991 \dots$

Solve Using the Square Root Property (6x+2)^2+4=28

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