Solve Using the Square Root Property (7x+12)^2=75

${(7 x + 12)}^{2} = 75$

Step 1

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

$7 x + 12 = \pm \sqrt{75}$

Step 2

Simplify $\pm \sqrt{75}$ .

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

Factor $25$ out of $75$ .

$7 x + 12 = \pm \sqrt{25 (3)}$

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

$7 x + 12 = \pm \sqrt{5^{2} \cdot 3}$

Pull terms out from under the radical.

$7 x + 12 = \pm 5 \sqrt{3}$

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.

$7 x + 12 = 5 \sqrt{3}$

Subtract $12$ from both sides of the equation.

$7 x = 5 \sqrt{3} - 12$

Divide each term in $7 x = 5 \sqrt{3} - 12$ by $7$ and simplify.

Divide each term in $7 x = 5 \sqrt{3} - 12$ by $7$ .

$\frac{7 x}{7} = \frac{5 \sqrt{3}}{7} + \frac{- 12}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 x}{7} = \frac{5 \sqrt{3}}{7} + \frac{- 12}{7}$

Divide $x$ by $1$ .

$x = \frac{5 \sqrt{3}}{7} + \frac{- 12}{7}$

Simplify the right side.

Move the negative in front of the fraction.

$x = \frac{5 \sqrt{3}}{7} - \frac{12}{7}$

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

$7 x + 12 = - 5 \sqrt{3}$

Subtract $12$ from both sides of the equation.

$7 x = - 5 \sqrt{3} - 12$

Divide each term in $7 x = - 5 \sqrt{3} - 12$ by $7$ and simplify.

Divide each term in $7 x = - 5 \sqrt{3} - 12$ by $7$ .

$\frac{7 x}{7} = \frac{- 5 \sqrt{3}}{7} + \frac{- 12}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 x}{7} = \frac{- 5 \sqrt{3}}{7} + \frac{- 12}{7}$

Divide $x$ by $1$ .

$x = \frac{- 5 \sqrt{3}}{7} + \frac{- 12}{7}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$x = - \frac{5 \sqrt{3}}{7} + \frac{- 12}{7}$

Move the negative in front of the fraction.

$x = - \frac{5 \sqrt{3}}{7} - \frac{12}{7}$

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

$x = \frac{5 \sqrt{3}}{7} - \frac{12}{7}, - \frac{5 \sqrt{3}}{7} - \frac{12}{7}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5 \sqrt{3}}{7} - \frac{12}{7}, - \frac{5 \sqrt{3}}{7} - \frac{12}{7}$

Decimal Form:

$x = - 0.47710656 \dots, - 2.95146486 \dots$

Solve Using the Square Root Property (7x+12)^2=75

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