Solve Using the Square Root Property (3x-1)^2=4/9

${(3 x - 1)}^{2} = \frac{4}{9}$

Step 1

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

$3 x - 1 = \pm \sqrt{\frac{4}{9}}$

Step 2

Simplify $\pm \sqrt{\frac{4}{9}}$ .

Rewrite $\sqrt{\frac{4}{9}}$ as $\frac{\sqrt{4}}{\sqrt{9}}$ .

$3 x - 1 = \pm \frac{\sqrt{4}}{\sqrt{9}}$

Simplify the numerator.

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

$3 x - 1 = \pm \frac{\sqrt{2^{2}}}{\sqrt{9}}$

Pull terms out from under the radical, assuming positive real numbers.

$3 x - 1 = \pm \frac{2}{\sqrt{9}}$

Simplify the denominator.

Rewrite $9$ as $3^{2}$ .

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

Pull terms out from under the radical, assuming positive real numbers.

$3 x - 1 = \pm \frac{2}{3}$

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.

$3 x - 1 = \frac{2}{3}$

Move all terms not containing $x$ to the right side of the equation.

Add $1$ to both sides of the equation.

$3 x = \frac{2}{3} + 1$

Write $1$ as a fraction with a common denominator.

$3 x = \frac{2}{3} + \frac{3}{3}$

Combine the numerators over the common denominator.

$3 x = \frac{2 + 3}{3}$

Add $2$ and $3$ .

$3 x = \frac{5}{3}$

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

Divide each term in $3 x = \frac{5}{3}$ by $3$ .

$\frac{3 x}{3} = \frac{\frac{5}{3}}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{\frac{5}{3}}{3}$

Divide $x$ by $1$ .

$x = \frac{\frac{5}{3}}{3}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{5}{3} \cdot \frac{1}{3}$

Multiply $\frac{5}{3} \cdot \frac{1}{3}$ .

Multiply $\frac{5}{3}$ by $\frac{1}{3}$ .

$x = \frac{5}{3 \cdot 3}$

Multiply $3$ by $3$ .

$x = \frac{5}{9}$

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

$3 x - 1 = - \frac{2}{3}$

Move all terms not containing $x$ to the right side of the equation.

Add $1$ to both sides of the equation.

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

Write $1$ as a fraction with a common denominator.

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

Combine the numerators over the common denominator.

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

Add $- 2$ and $3$ .

$3 x = \frac{1}{3}$

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

Divide each term in $3 x = \frac{1}{3}$ by $3$ .

$\frac{3 x}{3} = \frac{\frac{1}{3}}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{\frac{1}{3}}{3}$

Divide $x$ by $1$ .

$x = \frac{\frac{1}{3}}{3}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{1}{3} \cdot \frac{1}{3}$

Multiply $\frac{1}{3} \cdot \frac{1}{3}$ .

Multiply $\frac{1}{3}$ by $\frac{1}{3}$ .

$x = \frac{1}{3 \cdot 3}$

Multiply $3$ by $3$ .

$x = \frac{1}{9}$

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

$x = \frac{5}{9}, \frac{1}{9}$

Solve Using the Square Root Property (3x-1)^2=4/9

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