Algebra Examples

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

Step 1

Factor $2 x^{2} - 2$ .

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

Factor $2$ out of $- 2$ .

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

Step 1.1.3

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

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

Step 1.2

Rewrite $1$ as $1^{2}$ .

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

Step 1.3

Factor.

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Step 1.3.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

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

Step 1.3.2

Remove unnecessary parentheses.

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

Step 2

Factor $x$ out of $x^{2} + 9 x$ .

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Step 2.1

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

$f (x) = \frac{2 (x + 1) (x - 1)}{x \cdot x + 9 x}$

Step 2.2

Factor $x$ out of $9 x$ .

$f (x) = \frac{2 (x + 1) (x - 1)}{x \cdot x + x \cdot 9}$

Step 2.3

Factor $x$ out of $x \cdot x + x \cdot 9$ .

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

Step 3

Since no factors can be removed from the denominator, there are no holes in the graph.

No holes

Step 4