Pre-Algebra Examples

${(8 x - 5)}^{2} = - 9$

Step 1

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

$8 x - 5 = \pm \sqrt{- 9}$

Step 2

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

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Step 2.1

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

$8 x - 5 = \pm \sqrt{- 1 (9)}$

Step 2.2

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

$8 x - 5 = \pm \sqrt{- 1} \cdot \sqrt{9}$

Step 2.3

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

$8 x - 5 = \pm i \cdot \sqrt{9}$

Step 2.4

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

$8 x - 5 = \pm i \cdot \sqrt{3^{2}}$

Step 2.5

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

$8 x - 5 = \pm i \cdot 3$

Step 2.6

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

$8 x - 5 = \pm 3 i$

Step 3

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

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Step 3.1

First, use the positive value of the $\pm$ to find the first solution.

$8 x - 5 = 3 i$

Step 3.2

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

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Step 3.2.1

Add $5$ to both sides of the equation.

$8 x = 3 i + 5$

Step 3.2.2

Reorder $3 i$ and $5$ .

$8 x = 5 + 3 i$

Step 3.3

Divide each term in $8 x = 5 + 3 i$ by $8$ and simplify.

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Step 3.3.1

Divide each term in $8 x = 5 + 3 i$ by $8$ .

$\frac{8 x}{8} = \frac{5}{8} + \frac{3 i}{8}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $8$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{8 x}{8} = \frac{5}{8} + \frac{3 i}{8}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{5}{8} + \frac{3 i}{8}$

Step 3.4

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

$8 x - 5 = - 3 i$

Step 3.5

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

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Step 3.5.1

Add $5$ to both sides of the equation.

$8 x = - 3 i + 5$

Step 3.5.2

Reorder $- 3 i$ and $5$ .

$8 x = 5 - 3 i$

Step 3.6

Divide each term in $8 x = 5 - 3 i$ by $8$ and simplify.

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Step 3.6.1

Divide each term in $8 x = 5 - 3 i$ by $8$ .

$\frac{8 x}{8} = \frac{5}{8} + \frac{- 3 i}{8}$

Step 3.6.2

Simplify the left side.

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Step 3.6.2.1

Cancel the common factor of $8$ .

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Step 3.6.2.1.1

Cancel the common factor.

$\frac{8 x}{8} = \frac{5}{8} + \frac{- 3 i}{8}$

Step 3.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{5}{8} + \frac{- 3 i}{8}$

Step 3.6.3

Simplify the right side.

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Step 3.6.3.1

Move the negative in front of the fraction.

$x = \frac{5}{8} - \frac{3 i}{8}$

Step 3.7

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

$x = \frac{5}{8} + \frac{3 i}{8}, \frac{5}{8} - \frac{3 i}{8}$