Solve Using the Square Root Property (9x+25)^2=75

${(9 x + 25)}^{2} = 75$

Step 1

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

$9 x + 25 = \pm \sqrt{75}$

Step 2

Simplify $\pm \sqrt{75}$ .

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

Factor $25$ out of $75$ .

$9 x + 25 = \pm \sqrt{25 (3)}$

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

$9 x + 25 = \pm \sqrt{5^{2} \cdot 3}$

Pull terms out from under the radical.

$9 x + 25 = \pm 5 \sqrt{3}$

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.

$9 x + 25 = 5 \sqrt{3}$

Subtract $25$ from both sides of the equation.

$9 x = 5 \sqrt{3} - 25$

Divide each term in $9 x = 5 \sqrt{3} - 25$ by $9$ and simplify.

Divide each term in $9 x = 5 \sqrt{3} - 25$ by $9$ .

$\frac{9 x}{9} = \frac{5 \sqrt{3}}{9} + \frac{- 25}{9}$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 x}{9} = \frac{5 \sqrt{3}}{9} + \frac{- 25}{9}$

Divide $x$ by $1$ .

$x = \frac{5 \sqrt{3}}{9} + \frac{- 25}{9}$

Simplify the right side.

Move the negative in front of the fraction.

$x = \frac{5 \sqrt{3}}{9} - \frac{25}{9}$

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

$9 x + 25 = - 5 \sqrt{3}$

Subtract $25$ from both sides of the equation.

$9 x = - 5 \sqrt{3} - 25$

Divide each term in $9 x = - 5 \sqrt{3} - 25$ by $9$ and simplify.

Divide each term in $9 x = - 5 \sqrt{3} - 25$ by $9$ .

$\frac{9 x}{9} = \frac{- 5 \sqrt{3}}{9} + \frac{- 25}{9}$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 x}{9} = \frac{- 5 \sqrt{3}}{9} + \frac{- 25}{9}$

Divide $x$ by $1$ .

$x = \frac{- 5 \sqrt{3}}{9} + \frac{- 25}{9}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$x = - \frac{5 \sqrt{3}}{9} + \frac{- 25}{9}$

Move the negative in front of the fraction.

$x = - \frac{5 \sqrt{3}}{9} - \frac{25}{9}$

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

$x = \frac{5 \sqrt{3}}{9} - \frac{25}{9}, - \frac{5 \sqrt{3}}{9} - \frac{25}{9}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5 \sqrt{3}}{9} - \frac{25}{9}, - \frac{5 \sqrt{3}}{9} - \frac{25}{9}$

Decimal Form:

$x = - 1.81552732 \dots, - 3.74002822 \dots$

Solve Using the Square Root Property (9x+25)^2=75

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