Finite Math Examples

$\frac{x}{x^{2} - 9} + \frac{3}{x (x - 3)}$

Step 1

Simplify the denominator.

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Step 1.1

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

$\frac{x}{x^{2} - 3^{2}} + \frac{3}{x (x - 3)}$

Step 1.2

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 $b = 3$ .

$\frac{x}{(x + 3) (x - 3)} + \frac{3}{x (x - 3)}$

Step 2

To write $\frac{x}{(x + 3) (x - 3)}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 3

To write $\frac{3}{x (x - 3)}$ as a fraction with a common denominator, multiply by $\frac{x + 3}{x + 3}$ .

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

Step 4

Write each expression with a common denominator of $(x + 3) (x - 3) x$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Reorder the factors of $(x + 3) (x - 3) x$ .

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

Step 4.4

Reorder the factors of $x (x - 3) (x + 3)$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $x$ by $x$ .

$\frac{x^{2} + 3 (x + 3)}{x (x + 3) (x - 3)}$

Step 6.2

Apply the distributive property.

$\frac{x^{2} + 3 x + 3 \cdot 3}{x (x + 3) (x - 3)}$

Step 6.3

Multiply $3$ by $3$ .

$\frac{x^{2} + 3 x + 9}{x (x + 3) (x - 3)}$