Solve Using the Square Root Property (5x-3)^2=-3

${(5 x - 3)}^{2} = - 3$

Step 1

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

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

Step 2

Simplify $\pm \sqrt{- 3}$ .

Rewrite $- 3$ as $- 1 (3)$ .

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

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

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

Rewrite $\sqrt{- 1}$ as $i$ .

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

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.

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

Add $3$ to both sides of the equation.

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

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

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

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

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

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

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

Add $3$ to both sides of the equation.

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

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

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

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

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

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

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