Algebra Examples

$q (3) = \frac{1}{x^{2} - 9}$

Step 1

Divide each term in $q (3) = \frac{1}{x^{2} - 9}$ by $3$ and simplify.

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Step 1.1

Divide each term in $q (3) = \frac{1}{x^{2} - 9}$ by $3$ .

$\frac{q \cdot 3}{3} = \frac{\frac{1}{x^{2} - 9}}{3}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $3$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{q \cdot 3}{3} = \frac{\frac{1}{x^{2} - 9}}{3}$

Step 1.2.1.2

Divide $q$ by $1$ .

$q = \frac{\frac{1}{x^{2} - 9}}{3}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Multiply the numerator by the reciprocal of the denominator.

$q = \frac{1}{x^{2} - 9} \cdot \frac{1}{3}$

Step 1.3.2

Simplify the denominator.

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Step 1.3.2.1

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

$q = \frac{1}{x^{2} - 3^{2}} \cdot \frac{1}{3}$

Step 1.3.2.2

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 $b = 3$ .

$q = \frac{1}{(x + 3) (x - 3)} \cdot \frac{1}{3}$

Step 1.3.3

Combine fractions.

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Step 1.3.3.1

Multiply $\frac{1}{(x + 3) (x - 3)}$ by $\frac{1}{3}$ .

$q = \frac{1}{(x + 3) (x - 3) \cdot 3}$

Step 1.3.3.2

Move $3$ to the left of $(x + 3) (x - 3)$ .

$q = \frac{1}{3 (x + 3) (x - 3)}$

Step 2

To rewrite as a function of $x$ , write the equation so that $q$ is by itself on one side of the equal sign and an expression involving only $x$ is on the other side.

$f (x) = \frac{1}{3 (x + 3) (x - 3)}$