Solve Using the Square Root Property 3x^2-2=-77

$3 x^{2} - 2 = - 77$

Step 1

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

Add $2$ to both sides of the equation.

$3 x^{2} = - 77 + 2$

Add $- 77$ and $2$ .

$3 x^{2} = - 75$

Step 2

Divide each term in $3 x^{2} = - 75$ by $3$ and simplify.

Divide each term in $3 x^{2} = - 75$ by $3$ .

$\frac{3 x^{2}}{3} = \frac{- 75}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x^{2}}{3} = \frac{- 75}{3}$

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 75}{3}$

Simplify the right side.

Divide $- 75$ by $3$ .

$x^{2} = - 25$

Step 3

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

$x = \pm \sqrt{- 25}$

Step 4

Simplify $\pm \sqrt{- 25}$ .

Rewrite $- 25$ as $- 1 (25)$ .

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

Rewrite $\sqrt{- 1 (25)}$ as $\sqrt{- 1} \cdot \sqrt{25}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{25}$

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

$x = \pm i \cdot \sqrt{25}$

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

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

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

$x = \pm i \cdot 5$

Move $5$ to the left of $i$ .

$x = \pm 5 i$

Step 5

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.

$x = 5 i$

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

$x = - 5 i$

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

$x = 5 i, - 5 i$