Finite Math Examples

$\frac{9 x + 18}{x^{2} - 9} - \frac{x + 6}{x - 3}$

Step 1

Simplify each term.

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Step 1.1

Factor $9$ out of $9 x + 18$ .

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Step 1.1.1

Factor $9$ out of $9 x$ .

$\frac{9 (x) + 18}{x^{2} - 9} - \frac{x + 6}{x - 3}$

Step 1.1.2

Factor $9$ out of $18$ .

$\frac{9 x + 9 \cdot 2}{x^{2} - 9} - \frac{x + 6}{x - 3}$

Step 1.1.3

Factor $9$ out of $9 x + 9 \cdot 2$ .

$\frac{9 (x + 2)}{x^{2} - 9} - \frac{x + 6}{x - 3}$

Step 1.2

Simplify the denominator.

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Step 1.2.1

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

$\frac{9 (x + 2)}{x^{2} - 3^{2}} - \frac{x + 6}{x - 3}$

Step 1.2.2

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 $b = 3$ .

$\frac{9 (x + 2)}{(x + 3) (x - 3)} - \frac{x + 6}{x - 3}$

Step 2

To write $- \frac{x + 6}{x - 3}$ as a fraction with a common denominator, multiply by $\frac{x + 3}{x + 3}$ .

$\frac{9 (x + 2)}{(x + 3) (x - 3)} - \frac{x + 6}{x - 3} \cdot \frac{x + 3}{x + 3}$

Step 3

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

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Step 3.1

Multiply $\frac{x + 6}{x - 3}$ by $\frac{x + 3}{x + 3}$ .

$\frac{9 (x + 2)}{(x + 3) (x - 3)} - \frac{(x + 6) (x + 3)}{(x - 3) (x + 3)}$

Step 3.2

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

$\frac{9 (x + 2)}{(x + 3) (x - 3)} - \frac{(x + 6) (x + 3)}{(x + 3) (x - 3)}$

Step 4

Combine the numerators over the common denominator.

$\frac{9 (x + 2) - (x + 6) (x + 3)}{(x + 3) (x - 3)}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

$\frac{9 x + 9 \cdot 2 - (x + 6) (x + 3)}{(x + 3) (x - 3)}$

Step 5.2

Multiply $9$ by $2$ .

$\frac{9 x + 18 - (x + 6) (x + 3)}{(x + 3) (x - 3)}$

Step 5.3

Apply the distributive property.

$\frac{9 x + 18 + (- x - 1 \cdot 6) (x + 3)}{(x + 3) (x - 3)}$

Step 5.4

Multiply $- 1$ by $6$ .

$\frac{9 x + 18 + (- x - 6) (x + 3)}{(x + 3) (x - 3)}$

Step 5.5

Expand $(- x - 6) (x + 3)$ using the FOIL Method.

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Step 5.5.1

Apply the distributive property.

$\frac{9 x + 18 - x (x + 3) - 6 (x + 3)}{(x + 3) (x - 3)}$

Step 5.5.2

Apply the distributive property.

$\frac{9 x + 18 - x \cdot x - x \cdot 3 - 6 (x + 3)}{(x + 3) (x - 3)}$

Step 5.5.3

Apply the distributive property.

$\frac{9 x + 18 - x \cdot x - x \cdot 3 - 6 x - 6 \cdot 3}{(x + 3) (x - 3)}$

Step 5.6

Simplify and combine like terms.

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Step 5.6.1

Simplify each term.

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Step 5.6.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 5.6.1.1.1

Move $x$ .

$\frac{9 x + 18 - (x \cdot x) - x \cdot 3 - 6 x - 6 \cdot 3}{(x + 3) (x - 3)}$

Step 5.6.1.1.2

Multiply $x$ by $x$ .

$\frac{9 x + 18 - x^{2} - x \cdot 3 - 6 x - 6 \cdot 3}{(x + 3) (x - 3)}$

Step 5.6.1.2

Multiply $3$ by $- 1$ .

$\frac{9 x + 18 - x^{2} - 3 x - 6 x - 6 \cdot 3}{(x + 3) (x - 3)}$

Step 5.6.1.3

Multiply $- 6$ by $3$ .

$\frac{9 x + 18 - x^{2} - 3 x - 6 x - 18}{(x + 3) (x - 3)}$

Step 5.6.2

Subtract $6 x$ from $- 3 x$ .

$\frac{9 x + 18 - x^{2} - 9 x - 18}{(x + 3) (x - 3)}$

Step 5.7

Subtract $9 x$ from $9 x$ .

$\frac{18 - x^{2} + 0 - 18}{(x + 3) (x - 3)}$

Step 5.8

Add $18$ and $0$ .

$\frac{- x^{2} + 18 - 18}{(x + 3) (x - 3)}$

Step 5.9

Subtract $18$ from $18$ .

$\frac{- x^{2} + 0}{(x + 3) (x - 3)}$

Step 5.10

Add $- x^{2}$ and $0$ .

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

Step 6

Move the negative in front of the fraction.

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