Algebra Examples

${(5 x - 1)}^{2} = \frac{16}{25}$

Step 1

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

$5 x - 1 = \pm \sqrt{\frac{16}{25}}$

Step 2

Simplify $\pm \sqrt{\frac{16}{25}}$ .

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Step 2.1

Rewrite $\sqrt{\frac{16}{25}}$ as $\frac{\sqrt{16}}{\sqrt{25}}$ .

$5 x - 1 = \pm \frac{\sqrt{16}}{\sqrt{25}}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

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

$5 x - 1 = \pm \frac{\sqrt{4^{2}}}{\sqrt{25}}$

Step 2.2.2

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

$5 x - 1 = \pm \frac{4}{\sqrt{25}}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

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

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

Step 2.3.2

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

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

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.

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

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 $1$ to both sides of the equation.

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

Step 3.2.2

Write $1$ as a fraction with a common denominator.

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

Step 3.2.3

Combine the numerators over the common denominator.

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

Step 3.2.4

Add $4$ and $5$ .

$5 x = \frac{9}{5}$

Step 3.3

Divide each term in $5 x = \frac{9}{5}$ by $5$ and simplify.

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Step 3.3.1

Divide each term in $5 x = \frac{9}{5}$ by $5$ .

$\frac{5 x}{5} = \frac{\frac{9}{5}}{5}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $5$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{\frac{9}{5}}{5}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{9}{5}}{5}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{9}{5} \cdot \frac{1}{5}$

Step 3.3.3.2

Multiply $\frac{9}{5} \cdot \frac{1}{5}$ .

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Step 3.3.3.2.1

Multiply $\frac{9}{5}$ by $\frac{1}{5}$ .

$x = \frac{9}{5 \cdot 5}$

Step 3.3.3.2.2

Multiply $5$ by $5$ .

$x = \frac{9}{25}$

Step 3.4

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

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

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 $1$ to both sides of the equation.

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

Step 3.5.2

Write $1$ as a fraction with a common denominator.

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

Step 3.5.3

Combine the numerators over the common denominator.

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

Step 3.5.4

Add $- 4$ and $5$ .

$5 x = \frac{1}{5}$

Step 3.6

Divide each term in $5 x = \frac{1}{5}$ by $5$ and simplify.

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Step 3.6.1

Divide each term in $5 x = \frac{1}{5}$ by $5$ .

$\frac{5 x}{5} = \frac{\frac{1}{5}}{5}$

Step 3.6.2

Simplify the left side.

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Step 3.6.2.1

Cancel the common factor of $5$ .

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Step 3.6.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{\frac{1}{5}}{5}$

Step 3.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{1}{5}}{5}$

Step 3.6.3

Simplify the right side.

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Step 3.6.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{1}{5} \cdot \frac{1}{5}$

Step 3.6.3.2

Multiply $\frac{1}{5} \cdot \frac{1}{5}$ .

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Step 3.6.3.2.1

Multiply $\frac{1}{5}$ by $\frac{1}{5}$ .

$x = \frac{1}{5 \cdot 5}$

Step 3.6.3.2.2

Multiply $5$ by $5$ .

$x = \frac{1}{25}$

Step 3.7

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

$x = \frac{9}{25}, \frac{1}{25}$