Finite Math Examples

$\frac{7}{x^{2} - 9} + \frac{6}{x^{2} - 7 x + 12} - \frac{5}{x^{2} - x - 12}$

Step 1

Simplify each term.

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Step 1.1

Simplify the denominator.

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Step 1.1.1

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

$\frac{7}{x^{2} - 3^{2}} + \frac{6}{x^{2} - 7 x + 12} - \frac{5}{x^{2} - x - 12}$

Step 1.1.2

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 $b = 3$ .

$\frac{7}{(x + 3) (x - 3)} + \frac{6}{x^{2} - 7 x + 12} - \frac{5}{x^{2} - x - 12}$

Step 1.2

Factor $x^{2} - 7 x + 12$ using the AC method.

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Step 1.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $12$ and whose sum is $- 7$ .

$- 4, - 3$

Step 1.2.2

Write the factored form using these integers.

$\frac{7}{(x + 3) (x - 3)} + \frac{6}{(x - 4) (x - 3)} - \frac{5}{x^{2} - x - 12}$

Step 1.3

Factor $x^{2} - x - 12$ using the AC method.

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Step 1.3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 12$ and whose sum is $- 1$ .

$- 4, 3$

Step 1.3.2

Write the factored form using these integers.

$\frac{7}{(x + 3) (x - 3)} + \frac{6}{(x - 4) (x - 3)} - \frac{5}{(x - 4) (x + 3)}$

Step 2

Find the common denominator.

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Step 2.1

Multiply $\frac{7}{(x + 3) (x - 3)}$ by $\frac{x - 4}{x - 4}$ .

$\frac{7}{(x + 3) (x - 3)} \cdot \frac{x - 4}{x - 4} + \frac{6}{(x - 4) (x - 3)} - \frac{5}{(x - 4) (x + 3)}$

Step 2.2

Multiply $\frac{7}{(x + 3) (x - 3)}$ by $\frac{x - 4}{x - 4}$ .

$\frac{7 (x - 4)}{(x + 3) (x - 3) (x - 4)} + \frac{6}{(x - 4) (x - 3)} - \frac{5}{(x - 4) (x + 3)}$

Step 2.3

Multiply $\frac{6}{(x - 4) (x - 3)}$ by $\frac{x + 3}{x + 3}$ .

$\frac{7 (x - 4)}{(x + 3) (x - 3) (x - 4)} + \frac{6}{(x - 4) (x - 3)} \cdot \frac{x + 3}{x + 3} - \frac{5}{(x - 4) (x + 3)}$

Step 2.4

Multiply $\frac{6}{(x - 4) (x - 3)}$ by $\frac{x + 3}{x + 3}$ .

$\frac{7 (x - 4)}{(x + 3) (x - 3) (x - 4)} + \frac{6 (x + 3)}{(x - 4) (x - 3) (x + 3)} - \frac{5}{(x - 4) (x + 3)}$

Step 2.5

Multiply $\frac{5}{(x - 4) (x + 3)}$ by $\frac{x - 3}{x - 3}$ .

$\frac{7 (x - 4)}{(x + 3) (x - 3) (x - 4)} + \frac{6 (x + 3)}{(x - 4) (x - 3) (x + 3)} - (\frac{5}{(x - 4) (x + 3)} \cdot \frac{x - 3}{x - 3})$

Step 2.6

Multiply $\frac{5}{(x - 4) (x + 3)}$ by $\frac{x - 3}{x - 3}$ .

$\frac{7 (x - 4)}{(x + 3) (x - 3) (x - 4)} + \frac{6 (x + 3)}{(x - 4) (x - 3) (x + 3)} - \frac{5 (x - 3)}{(x - 4) (x + 3) (x - 3)}$

Step 2.7

Reorder the factors of $(x - 4) (x - 3) (x + 3)$ .

$\frac{7 (x - 4)}{(x + 3) (x - 3) (x - 4)} + \frac{6 (x + 3)}{(x + 3) (x - 3) (x - 4)} - \frac{5 (x - 3)}{(x - 4) (x + 3) (x - 3)}$

Step 2.8

Reorder the factors of $(x - 4) (x + 3) (x - 3)$ .

$\frac{7 (x - 4)}{(x + 3) (x - 3) (x - 4)} + \frac{6 (x + 3)}{(x + 3) (x - 3) (x - 4)} - \frac{5 (x - 3)}{(x + 3) (x - 3) (x - 4)}$

Step 3

Combine the numerators over the common denominator.

$\frac{7 (x - 4) + 6 (x + 3) - 5 (x - 3)}{(x + 3) (x - 3) (x - 4)}$

Step 4

Simplify each term.

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Step 4.1

Apply the distributive property.

$\frac{7 x + 7 \cdot - 4 + 6 (x + 3) - 5 (x - 3)}{(x + 3) (x - 3) (x - 4)}$

Step 4.2

Multiply $7$ by $- 4$ .

$\frac{7 x - 28 + 6 (x + 3) - 5 (x - 3)}{(x + 3) (x - 3) (x - 4)}$

Step 4.3

Apply the distributive property.

$\frac{7 x - 28 + 6 x + 6 \cdot 3 - 5 (x - 3)}{(x + 3) (x - 3) (x - 4)}$

Step 4.4

Multiply $6$ by $3$ .

$\frac{7 x - 28 + 6 x + 18 - 5 (x - 3)}{(x + 3) (x - 3) (x - 4)}$

Step 4.5

Apply the distributive property.

$\frac{7 x - 28 + 6 x + 18 - 5 x - 5 \cdot - 3}{(x + 3) (x - 3) (x - 4)}$

Step 4.6

Multiply $- 5$ by $- 3$ .

$\frac{7 x - 28 + 6 x + 18 - 5 x + 15}{(x + 3) (x - 3) (x - 4)}$

Step 5

Simplify by adding terms.

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Step 5.1

Add $7 x$ and $6 x$ .

$\frac{13 x - 28 + 18 - 5 x + 15}{(x + 3) (x - 3) (x - 4)}$

Step 5.2

Subtract $5 x$ from $13 x$ .

$\frac{8 x - 28 + 18 + 15}{(x + 3) (x - 3) (x - 4)}$

Step 5.3

Simplify by adding numbers.

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Step 5.3.1

Add $- 28$ and $18$ .

$\frac{8 x - 10 + 15}{(x + 3) (x - 3) (x - 4)}$

Step 5.3.2

Add $- 10$ and $15$ .

$\frac{8 x + 5}{(x + 3) (x - 3) (x - 4)}$