Algebra Examples

${(x - 7)}^{2} = \frac{64}{49}$

Step 1

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

$x - 7 = \pm \sqrt{\frac{64}{49}}$

Step 2

Simplify $\pm \sqrt{\frac{64}{49}}$ .

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Step 2.1

Rewrite $\sqrt{\frac{64}{49}}$ as $\frac{\sqrt{64}}{\sqrt{49}}$ .

$x - 7 = \pm \frac{\sqrt{64}}{\sqrt{49}}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

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

$x - 7 = \pm \frac{\sqrt{8^{2}}}{\sqrt{49}}$

Step 2.2.2

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

$x - 7 = \pm \frac{8}{\sqrt{49}}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Rewrite $49$ as $7^{2}$ .

$x - 7 = \pm \frac{8}{\sqrt{7^{2}}}$

Step 2.3.2

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

$x - 7 = \pm \frac{8}{7}$

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.

$x - 7 = \frac{8}{7}$

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 $7$ to both sides of the equation.

$x = \frac{8}{7} + 7$

Step 3.2.2

To write $7$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$x = \frac{8}{7} + 7 \cdot \frac{7}{7}$

Step 3.2.3

Combine $7$ and $\frac{7}{7}$ .

$x = \frac{8}{7} + \frac{7 \cdot 7}{7}$

Step 3.2.4

Combine the numerators over the common denominator.

$x = \frac{8 + 7 \cdot 7}{7}$

Step 3.2.5

Simplify the numerator.

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Step 3.2.5.1

Multiply $7$ by $7$ .

$x = \frac{8 + 49}{7}$

Step 3.2.5.2

Add $8$ and $49$ .

$x = \frac{57}{7}$

Step 3.3

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

$x - 7 = - \frac{8}{7}$

Step 3.4

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

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Step 3.4.1

Add $7$ to both sides of the equation.

$x = - \frac{8}{7} + 7$

Step 3.4.2

To write $7$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$x = - \frac{8}{7} + 7 \cdot \frac{7}{7}$

Step 3.4.3

Combine $7$ and $\frac{7}{7}$ .

$x = - \frac{8}{7} + \frac{7 \cdot 7}{7}$

Step 3.4.4

Combine the numerators over the common denominator.

$x = \frac{- 8 + 7 \cdot 7}{7}$

Step 3.4.5

Simplify the numerator.

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Step 3.4.5.1

Multiply $7$ by $7$ .

$x = \frac{- 8 + 49}{7}$

Step 3.4.5.2

Add $- 8$ and $49$ .

$x = \frac{41}{7}$

Step 3.5

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

$x = \frac{57}{7}, \frac{41}{7}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{57}{7}, \frac{41}{7}$

Decimal Form:

$x = 8.\overline{142857}, 5.\overline{857142}$

Mixed Number Form:

$x = 8 \frac{1}{7}, 5 \frac{6}{7}$