Pre-Algebra Examples

$\frac{x^{4} - 2}{x^{2} - 2} + \frac{2}{- x^{2} + 2}$

Step 1

Simplify terms.

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Step 1.1

Factor $- 1$ out of $- x^{2}$ .

$\frac{x^{4} - 2}{x^{2} - 2} + \frac{2}{- (x^{2}) + 2}$

Step 1.2

Rewrite $2$ as $- 1 (- 2)$ .

$\frac{x^{4} - 2}{x^{2} - 2} + \frac{2}{- (x^{2}) - 1 (- 2)}$

Step 1.3

Factor $- 1$ out of $- (x^{2}) - 1 (- 2)$ .

$\frac{x^{4} - 2}{x^{2} - 2} + \frac{2}{- (x^{2} - 2)}$

Step 1.4

Move a negative from the denominator of $\frac{2}{- (x^{2} - 2)}$ to the numerator.

$\frac{x^{4} - 2}{x^{2} - 2} + \frac{- 1 \cdot 2}{x^{2} - 2}$

Step 1.5

Combine the numerators over the common denominator.

$\frac{x^{4} - 2 - 1 \cdot 2}{x^{2} - 2}$

Step 2

Simplify the numerator.

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Step 2.1

Multiply $- 1$ by $2$ .

$\frac{x^{4} - 2 - 2}{x^{2} - 2}$

Step 2.2

Subtract $2$ from $- 2$ .

$\frac{x^{4} - 4}{x^{2} - 2}$

Step 2.3

Rewrite $x^{4} - 4$ in a factored form.

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Step 2.3.1

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$\frac{{(x^{2})}^{2} - 4}{x^{2} - 2}$

Step 2.3.2

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

$\frac{{(x^{2})}^{2} - 2^{2}}{x^{2} - 2}$

Step 2.3.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{2}$ and $b = 2$ .

$\frac{(x^{2} + 2) (x^{2} - 2)}{x^{2} - 2}$

Step 3

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

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Step 3.1

Cancel the common factor.

$\frac{(x^{2} + 2) (x^{2} - 2)}{x^{2} - 2}$

Step 3.2

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

$x^{2} + 2$