Solve Using the Square Root Property (3x+2)^2+64=0

${(3 x + 2)}^{2} + 64 = 0$

Step 1

Subtract $64$ from both sides of the equation.

${(3 x + 2)}^{2} = - 64$

Step 2

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

$3 x + 2 = \pm \sqrt{- 64}$

Step 3

Simplify $\pm \sqrt{- 64}$ .

Rewrite $- 64$ as $- 1 (64)$ .

$3 x + 2 = \pm \sqrt{- 1 (64)}$

Rewrite $\sqrt{- 1 (64)}$ as $\sqrt{- 1} \cdot \sqrt{64}$ .

$3 x + 2 = \pm \sqrt{- 1} \cdot \sqrt{64}$

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

$3 x + 2 = \pm i \cdot \sqrt{64}$

Rewrite $64$ as $8^{2}$ .

$3 x + 2 = \pm i \cdot \sqrt{8^{2}}$

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

$3 x + 2 = \pm i \cdot 8$

Move $8$ to the left of $i$ .

$3 x + 2 = \pm 8 i$

Step 4

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.

$3 x + 2 = 8 i$

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

Subtract $2$ from both sides of the equation.

$3 x = 8 i - 2$

Reorder $8 i$ and $- 2$ .

$3 x = - 2 + 8 i$

Divide each term in $3 x = - 2 + 8 i$ by $3$ and simplify.

Divide each term in $3 x = - 2 + 8 i$ by $3$ .

$\frac{3 x}{3} = \frac{- 2}{3} + \frac{8 i}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 2}{3} + \frac{8 i}{3}$

Divide $x$ by $1$ .

$x = \frac{- 2}{3} + \frac{8 i}{3}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{2}{3} + \frac{8 i}{3}$

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

$3 x + 2 = - 8 i$

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

Subtract $2$ from both sides of the equation.

$3 x = - 8 i - 2$

Reorder $- 8 i$ and $- 2$ .

$3 x = - 2 - 8 i$

Divide each term in $3 x = - 2 - 8 i$ by $3$ and simplify.

Divide each term in $3 x = - 2 - 8 i$ by $3$ .

$\frac{3 x}{3} = \frac{- 2}{3} + \frac{- 8 i}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 2}{3} + \frac{- 8 i}{3}$

Divide $x$ by $1$ .

$x = \frac{- 2}{3} + \frac{- 8 i}{3}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$x = - \frac{2}{3} + \frac{- 8 i}{3}$

Move the negative in front of the fraction.

$x = - \frac{2}{3} - \frac{8 i}{3}$

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

$x = - \frac{2}{3} + \frac{8 i}{3}, - \frac{2}{3} - \frac{8 i}{3}$

Solve Using the Square Root Property (3x+2)^2+64=0

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