Solve Using the Square Root Property (5x-2)^2=-9

${(5 x - 2)}^{2} = - 9$

Step 1

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

$5 x - 2 = \pm \sqrt{- 9}$

Step 2

Simplify $\pm \sqrt{- 9}$ .

Rewrite $- 9$ as $- 1 (9)$ .

$5 x - 2 = \pm \sqrt{- 1 (9)}$

Rewrite $\sqrt{- 1 (9)}$ as $\sqrt{- 1} \cdot \sqrt{9}$ .

$5 x - 2 = \pm \sqrt{- 1} \cdot \sqrt{9}$

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

$5 x - 2 = \pm i \cdot \sqrt{9}$

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

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

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

$5 x - 2 = \pm i \cdot 3$

Move $3$ to the left of $i$ .

$5 x - 2 = \pm 3 i$

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.

$5 x - 2 = 3 i$

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

Add $2$ to both sides of the equation.

$5 x = 3 i + 2$

Reorder $3 i$ and $2$ .

$5 x = 2 + 3 i$

Divide each term in $5 x = 2 + 3 i$ by $5$ and simplify.

Divide each term in $5 x = 2 + 3 i$ by $5$ .

$\frac{5 x}{5} = \frac{2}{5} + \frac{3 i}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} = \frac{2}{5} + \frac{3 i}{5}$

Divide $x$ by $1$ .

$x = \frac{2}{5} + \frac{3 i}{5}$

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

$5 x - 2 = - 3 i$

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

Add $2$ to both sides of the equation.

$5 x = - 3 i + 2$

Reorder $- 3 i$ and $2$ .

$5 x = 2 - 3 i$

Divide each term in $5 x = 2 - 3 i$ by $5$ and simplify.

Divide each term in $5 x = 2 - 3 i$ by $5$ .

$\frac{5 x}{5} = \frac{2}{5} + \frac{- 3 i}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} = \frac{2}{5} + \frac{- 3 i}{5}$

Divide $x$ by $1$ .

$x = \frac{2}{5} + \frac{- 3 i}{5}$

Simplify the right side.

Move the negative in front of the fraction.

$x = \frac{2}{5} - \frac{3 i}{5}$

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

$x = \frac{2}{5} + \frac{3 i}{5}, \frac{2}{5} - \frac{3 i}{5}$

Solve Using the Square Root Property (5x-2)^2=-9

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