Algebra Examples

$2 = \frac{20}{x^{2} + 2}$

Step 1

Rewrite the equation as $\frac{20}{x^{2} + 2} = 2$ .

$\frac{20}{x^{2} + 2} = 2$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x^{2} + 2, 1$

Step 2.2

Remove parentheses.

$x^{2} + 2, 1$

Step 2.3

The LCM of one and any expression is the expression.

$x^{2} + 2$

Step 3

Multiply each term in $\frac{20}{x^{2} + 2} = 2$ by $x^{2} + 2$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{20}{x^{2} + 2} = 2$ by $x^{2} + 2$ .

$\frac{20}{x^{2} + 2} (x^{2} + 2) = 2 (x^{2} + 2)$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $x^{2} + 2$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{20}{x^{2} + 2} (x^{2} + 2) = 2 (x^{2} + 2)$

Step 3.2.1.2

Rewrite the expression.

$20 = 2 (x^{2} + 2)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Apply the distributive property.

$20 = 2 x^{2} + 2 \cdot 2$

Step 3.3.2

Multiply $2$ by $2$ .

$20 = 2 x^{2} + 4$

Step 4

Solve the equation.

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Step 4.1

Rewrite the equation as $2 x^{2} + 4 = 20$ .

$2 x^{2} + 4 = 20$

Step 4.2

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

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Step 4.2.1

Subtract $4$ from both sides of the equation.

$2 x^{2} = 20 - 4$

Step 4.2.2

Subtract $4$ from $20$ .

$2 x^{2} = 16$

Step 4.3

Divide each term in $2 x^{2} = 16$ by $2$ and simplify.

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Step 4.3.1

Divide each term in $2 x^{2} = 16$ by $2$ .

$\frac{2 x^{2}}{2} = \frac{16}{2}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $2$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{2 x^{2}}{2} = \frac{16}{2}$

Step 4.3.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{16}{2}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Divide $16$ by $2$ .

$x^{2} = 8$

Step 4.4

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

$x = \pm \sqrt{8}$

Step 4.5

Simplify $\pm \sqrt{8}$ .

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Step 4.5.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 4.5.1.1

Factor $4$ out of $8$ .

$x = \pm \sqrt{4 (2)}$

Step 4.5.1.2

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

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

Step 4.5.2

Pull terms out from under the radical.

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

Step 4.6

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

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Step 4.6.1

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

$x = 2 \sqrt{2}$

Step 4.6.2

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

$x = - 2 \sqrt{2}$

Step 4.6.3

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

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 2.82842712 \dots, - 2.82842712 \dots$