Algebra Examples

${(5 x - 1)}^{2} + 16 = 0$

Step 1

Subtract $16$ from both sides of the equation.

${(5 x - 1)}^{2} = - 16$

Step 2

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

$5 x - 1 = \pm \sqrt{- 16}$

Step 3

Simplify $\pm \sqrt{- 16}$ .

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Step 3.1

Rewrite $- 16$ as $- 1 (16)$ .

$5 x - 1 = \pm \sqrt{- 1 (16)}$

Step 3.2

Rewrite $\sqrt{- 1 (16)}$ as $\sqrt{- 1} \cdot \sqrt{16}$ .

$5 x - 1 = \pm \sqrt{- 1} \cdot \sqrt{16}$

Step 3.3

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

$5 x - 1 = \pm i \cdot \sqrt{16}$

Step 3.4

Rewrite $16$ as $4^{2}$ .

$5 x - 1 = \pm i \cdot \sqrt{4^{2}}$

Step 3.5

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

$5 x - 1 = \pm i \cdot 4$

Step 3.6

Move $4$ to the left of $i$ .

$5 x - 1 = \pm 4 i$

Step 4

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

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Step 4.1

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

$5 x - 1 = 4 i$

Step 4.2

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

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Step 4.2.1

Add $1$ to both sides of the equation.

$5 x = 4 i + 1$

Step 4.2.2

Reorder $4 i$ and $1$ .

$5 x = 1 + 4 i$

Step 4.3

Divide each term in $5 x = 1 + 4 i$ by $5$ and simplify.

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Step 4.3.1

Divide each term in $5 x = 1 + 4 i$ by $5$ .

$\frac{5 x}{5} = \frac{1}{5} + \frac{4 i}{5}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $5$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{1}{5} + \frac{4 i}{5}$

Step 4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{5} + \frac{4 i}{5}$

Step 4.4

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

$5 x - 1 = - 4 i$

Step 4.5

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

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Step 4.5.1

Add $1$ to both sides of the equation.

$5 x = - 4 i + 1$

Step 4.5.2

Reorder $- 4 i$ and $1$ .

$5 x = 1 - 4 i$

Step 4.6

Divide each term in $5 x = 1 - 4 i$ by $5$ and simplify.

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Step 4.6.1

Divide each term in $5 x = 1 - 4 i$ by $5$ .

$\frac{5 x}{5} = \frac{1}{5} + \frac{- 4 i}{5}$

Step 4.6.2

Simplify the left side.

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Step 4.6.2.1

Cancel the common factor of $5$ .

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Step 4.6.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{1}{5} + \frac{- 4 i}{5}$

Step 4.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{5} + \frac{- 4 i}{5}$

Step 4.6.3

Simplify the right side.

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Step 4.6.3.1

Move the negative in front of the fraction.

$x = \frac{1}{5} - \frac{4 i}{5}$

Step 4.7

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

$x = \frac{1}{5} + \frac{4 i}{5}, \frac{1}{5} - \frac{4 i}{5}$