Solve Using the Square Root Property (3x-1)^2=75

${(3 x - 1)}^{2} = 75$

Step 1

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

$3 x - 1 = \pm \sqrt{75}$

Step 2

Simplify $\pm \sqrt{75}$ .

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

Factor $25$ out of $75$ .

$3 x - 1 = \pm \sqrt{25 (3)}$

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

$3 x - 1 = \pm \sqrt{5^{2} \cdot 3}$

Pull terms out from under the radical.

$3 x - 1 = \pm 5 \sqrt{3}$

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.

$3 x - 1 = 5 \sqrt{3}$

Add $1$ to both sides of the equation.

$3 x = 5 \sqrt{3} + 1$

Divide each term in $3 x = 5 \sqrt{3} + 1$ by $3$ and simplify.

Divide each term in $3 x = 5 \sqrt{3} + 1$ by $3$ .

$\frac{3 x}{3} = \frac{5 \sqrt{3}}{3} + \frac{1}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{5 \sqrt{3}}{3} + \frac{1}{3}$

Divide $x$ by $1$ .

$x = \frac{5 \sqrt{3}}{3} + \frac{1}{3}$

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

$3 x - 1 = - 5 \sqrt{3}$

Add $1$ to both sides of the equation.

$3 x = - 5 \sqrt{3} + 1$

Divide each term in $3 x = - 5 \sqrt{3} + 1$ by $3$ and simplify.

Divide each term in $3 x = - 5 \sqrt{3} + 1$ by $3$ .

$\frac{3 x}{3} = \frac{- 5 \sqrt{3}}{3} + \frac{1}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 5 \sqrt{3}}{3} + \frac{1}{3}$

Divide $x$ by $1$ .

$x = \frac{- 5 \sqrt{3}}{3} + \frac{1}{3}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{5 \sqrt{3}}{3} + \frac{1}{3}$

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

$x = \frac{5 \sqrt{3}}{3} + \frac{1}{3}, - \frac{5 \sqrt{3}}{3} + \frac{1}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5 \sqrt{3}}{3} + \frac{1}{3}, - \frac{5 \sqrt{3}}{3} + \frac{1}{3}$

Decimal Form:

$x = 3.22008467 \dots, - 2.55341801 \dots$

Solve Using the Square Root Property (3x-1)^2=75

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