Solve Using the Square Root Property (6x+16)^2=75

${(6 x + 16)}^{2} = 75$

Step 1

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

$6 x + 16 = \pm \sqrt{75}$

Step 2

Simplify $\pm \sqrt{75}$ .

Rewrite $75$ as $5^{2} \cdot 3$ .

Factor $25$ out of $75$ .

$6 x + 16 = \pm \sqrt{25 (3)}$

Rewrite $25$ as $5^{2}$ .

$6 x + 16 = \pm \sqrt{5^{2} \cdot 3}$

Pull terms out from under the radical.

$6 x + 16 = \pm 5 \sqrt{3}$

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.

$6 x + 16 = 5 \sqrt{3}$

Subtract $16$ from both sides of the equation.

$6 x = 5 \sqrt{3} - 16$

Divide each term in $6 x = 5 \sqrt{3} - 16$ by $6$ and simplify.

Divide each term in $6 x = 5 \sqrt{3} - 16$ by $6$ .

$\frac{6 x}{6} = \frac{5 \sqrt{3}}{6} + \frac{- 16}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 x}{6} = \frac{5 \sqrt{3}}{6} + \frac{- 16}{6}$

Divide $x$ by $1$ .

$x = \frac{5 \sqrt{3}}{6} + \frac{- 16}{6}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 16$ and $6$ .

Factor $2$ out of $- 16$ .

$x = \frac{5 \sqrt{3}}{6} + \frac{2 (- 8)}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

$x = \frac{5 \sqrt{3}}{6} + \frac{2 \cdot - 8}{2 \cdot 3}$

Cancel the common factor.

$x = \frac{5 \sqrt{3}}{6} + \frac{2 \cdot - 8}{2 \cdot 3}$

Rewrite the expression.

$x = \frac{5 \sqrt{3}}{6} + \frac{- 8}{3}$

Move the negative in front of the fraction.

$x = \frac{5 \sqrt{3}}{6} - \frac{8}{3}$

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

$6 x + 16 = - 5 \sqrt{3}$

Subtract $16$ from both sides of the equation.

$6 x = - 5 \sqrt{3} - 16$

Divide each term in $6 x = - 5 \sqrt{3} - 16$ by $6$ and simplify.

Divide each term in $6 x = - 5 \sqrt{3} - 16$ by $6$ .

$\frac{6 x}{6} = \frac{- 5 \sqrt{3}}{6} + \frac{- 16}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 x}{6} = \frac{- 5 \sqrt{3}}{6} + \frac{- 16}{6}$

Divide $x$ by $1$ .

$x = \frac{- 5 \sqrt{3}}{6} + \frac{- 16}{6}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$x = - \frac{5 \sqrt{3}}{6} + \frac{- 16}{6}$

Cancel the common factor of $- 16$ and $6$ .

Factor $2$ out of $- 16$ .

$x = - \frac{5 \sqrt{3}}{6} + \frac{2 (- 8)}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

$x = - \frac{5 \sqrt{3}}{6} + \frac{2 \cdot - 8}{2 \cdot 3}$

Cancel the common factor.

$x = - \frac{5 \sqrt{3}}{6} + \frac{2 \cdot - 8}{2 \cdot 3}$

Rewrite the expression.

$x = - \frac{5 \sqrt{3}}{6} + \frac{- 8}{3}$

Move the negative in front of the fraction.

$x = - \frac{5 \sqrt{3}}{6} - \frac{8}{3}$

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

$x = \frac{5 \sqrt{3}}{6} - \frac{8}{3}, - \frac{5 \sqrt{3}}{6} - \frac{8}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5 \sqrt{3}}{6} - \frac{8}{3}, - \frac{5 \sqrt{3}}{6} - \frac{8}{3}$

Decimal Form:

$x = - 1.22329099 \dots, - 4.11004233 \dots$

Solve Using the Square Root Property (6x+16)^2=75

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