Calculus Examples

$\frac{2}{{(x + 2)}^{2}} = \frac{1}{9}$

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.

$2 \cdot 9 = {(x + 2)}^{2} \cdot 1$

Step 2

Solve the equation for $x$ .

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Step 2.1

Rewrite the equation as ${(x + 2)}^{2} \cdot 1 = 2 \cdot 9$ .

${(x + 2)}^{2} \cdot 1 = 2 \cdot 9$

Step 2.2

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

${(x + 2)}^{2} = 2 \cdot 9$

Step 2.3

Multiply $2$ by $9$ .

${(x + 2)}^{2} = 18$

Step 2.4

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

$x + 2 = \pm \sqrt{18}$

Step 2.5

Simplify $\pm \sqrt{18}$ .

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Step 2.5.1

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 2.5.1.1

Factor $9$ out of $18$ .

$x + 2 = \pm \sqrt{9 (2)}$

Step 2.5.1.2

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

$x + 2 = \pm \sqrt{3^{2} \cdot 2}$

Step 2.5.2

Pull terms out from under the radical.

$x + 2 = \pm 3 \sqrt{2}$

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.

$x + 2 = 3 \sqrt{2}$

Step 2.6.2

Subtract $2$ from both sides of the equation.

$x = 3 \sqrt{2} - 2$

Step 2.6.3

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

$x + 2 = - 3 \sqrt{2}$

Step 2.6.4

Subtract $2$ from both sides of the equation.

$x = - 3 \sqrt{2} - 2$

Step 2.6.5

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

$x = 3 \sqrt{2} - 2, - 3 \sqrt{2} - 2$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = 3 \sqrt{2} - 2, - 3 \sqrt{2} - 2$

Decimal Form:

$x = 2.24264068 \dots, - 6.24264068 \dots$