Calculus Examples

$\frac{7}{{(x + 7)}^{2}} = \frac{169}{24}$

Step 1

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

$7 \cdot 24 = {(x + 7)}^{2} \cdot 169$

Step 2

Solve the equation for $x$ .

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Step 2.1

Rewrite the equation as ${(x + 7)}^{2} \cdot 169 = 7 \cdot 24$ .

${(x + 7)}^{2} \cdot 169 = 7 \cdot 24$

Step 2.2

Multiply $7$ by $24$ .

${(x + 7)}^{2} \cdot 169 = 168$

Step 2.3

Divide each term in ${(x + 7)}^{2} \cdot 169 = 168$ by $169$ and simplify.

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Step 2.3.1

Divide each term in ${(x + 7)}^{2} \cdot 169 = 168$ by $169$ .

$\frac{{(x + 7)}^{2} \cdot 169}{169} = \frac{168}{169}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $169$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{{(x + 7)}^{2} \cdot 169}{169} = \frac{168}{169}$

Step 2.3.2.1.2

Divide ${(x + 7)}^{2}$ by $1$ .

${(x + 7)}^{2} = \frac{168}{169}$

Step 2.4

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

$x + 7 = \pm \sqrt{\frac{168}{169}}$

Step 2.5

Simplify $\pm \sqrt{\frac{168}{169}}$ .

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Step 2.5.1

Rewrite $\sqrt{\frac{168}{169}}$ as $\frac{\sqrt{168}}{\sqrt{169}}$ .

$x + 7 = \pm \frac{\sqrt{168}}{\sqrt{169}}$

Step 2.5.2

Simplify the numerator.

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Step 2.5.2.1

Rewrite $168$ as $2^{2} \cdot 42$ .

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Step 2.5.2.1.1

Factor $4$ out of $168$ .

$x + 7 = \pm \frac{\sqrt{4 (42)}}{\sqrt{169}}$

Step 2.5.2.1.2

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

$x + 7 = \pm \frac{\sqrt{2^{2} \cdot 42}}{\sqrt{169}}$

Step 2.5.2.2

Pull terms out from under the radical.

$x + 7 = \pm \frac{2 \sqrt{42}}{\sqrt{169}}$

Step 2.5.3

Simplify the denominator.

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Step 2.5.3.1

Rewrite $169$ as $13^{2}$ .

$x + 7 = \pm \frac{2 \sqrt{42}}{\sqrt{13^{2}}}$

Step 2.5.3.2

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

$x + 7 = \pm \frac{2 \sqrt{42}}{13}$

Step 2.6

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

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Step 2.6.1

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

$x + 7 = \frac{2 \sqrt{42}}{13}$

Step 2.6.2

Subtract $7$ from both sides of the equation.

$x = \frac{2 \sqrt{42}}{13} - 7$

Step 2.6.3

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

$x + 7 = - \frac{2 \sqrt{42}}{13}$

Step 2.6.4

Subtract $7$ from both sides of the equation.

$x = - \frac{2 \sqrt{42}}{13} - 7$

Step 2.6.5

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

$x = \frac{2 \sqrt{42}}{13} - 7, - \frac{2 \sqrt{42}}{13} - 7$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{2 \sqrt{42}}{13} - 7, - \frac{2 \sqrt{42}}{13} - 7$

Decimal Form:

$x = - 6.00296296 \dots, - 7.99703703 \dots$