Solve Using the Square Root Property (3x-4)^2=42

${(3 x - 4)}^{2} = 42$

Step 1

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

$3 x - 4 = \pm \sqrt{42}$

Step 2

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.

$3 x - 4 = \sqrt{42}$

Add $4$ to both sides of the equation.

$3 x = \sqrt{42} + 4$

Divide each term in $3 x = \sqrt{42} + 4$ by $3$ and simplify.

Divide each term in $3 x = \sqrt{42} + 4$ by $3$ .

$\frac{3 x}{3} = \frac{\sqrt{42}}{3} + \frac{4}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{\sqrt{42}}{3} + \frac{4}{3}$

Divide $x$ by $1$ .

$x = \frac{\sqrt{42}}{3} + \frac{4}{3}$

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

$3 x - 4 = - \sqrt{42}$

Add $4$ to both sides of the equation.

$3 x = - \sqrt{42} + 4$

Divide each term in $3 x = - \sqrt{42} + 4$ by $3$ and simplify.

Divide each term in $3 x = - \sqrt{42} + 4$ by $3$ .

$\frac{3 x}{3} = \frac{- \sqrt{42}}{3} + \frac{4}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{- \sqrt{42}}{3} + \frac{4}{3}$

Divide $x$ by $1$ .

$x = \frac{- \sqrt{42}}{3} + \frac{4}{3}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{\sqrt{42}}{3} + \frac{4}{3}$

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

$x = \frac{\sqrt{42}}{3} + \frac{4}{3}, - \frac{\sqrt{42}}{3} + \frac{4}{3}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\sqrt{42}}{3} + \frac{4}{3}, - \frac{\sqrt{42}}{3} + \frac{4}{3}$

Decimal Form:

$x = 3.49358023 \dots, - 0.82691356 \dots$