Finite Math Examples

$\frac{8}{x^{2} - 13 x - 30} - \frac{3}{x + 2}$

Step 1

Factor $x^{2} - 13 x - 30$ using the AC method.

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Step 1.1

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 $- 30$ and whose sum is $- 13$ .

$- 15, 2$

Step 1.2

Write the factored form using these integers.

$\frac{8}{(x - 15) (x + 2)} - \frac{3}{x + 2}$

Step 2

To write $- \frac{3}{x + 2}$ as a fraction with a common denominator, multiply by $\frac{x - 15}{x - 15}$ .

$\frac{8}{(x - 15) (x + 2)} - \frac{3}{x + 2} \cdot \frac{x - 15}{x - 15}$

Step 3

Write each expression with a common denominator of $(x - 15) (x + 2)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{3}{x + 2}$ by $\frac{x - 15}{x - 15}$ .

$\frac{8}{(x - 15) (x + 2)} - \frac{3 (x - 15)}{(x + 2) (x - 15)}$

Step 3.2

Reorder the factors of $(x - 15) (x + 2)$ .

$\frac{8}{(x + 2) (x - 15)} - \frac{3 (x - 15)}{(x + 2) (x - 15)}$

Step 4

Combine the numerators over the common denominator.

$\frac{8 - 3 (x - 15)}{(x + 2) (x - 15)}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

$\frac{8 - 3 x - 3 \cdot - 15}{(x + 2) (x - 15)}$

Step 5.2

Multiply $- 3$ by $- 15$ .

$\frac{8 - 3 x + 45}{(x + 2) (x - 15)}$

Step 5.3

Add $8$ and $45$ .

$\frac{- 3 x + 53}{(x + 2) (x - 15)}$

Step 6

Simplify with factoring out.

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Step 6.1

Factor $- 1$ out of $- 3 x$ .

$\frac{- (3 x) + 53}{(x + 2) (x - 15)}$

Step 6.2

Rewrite $53$ as $- 1 (- 53)$ .

$\frac{- (3 x) - 1 (- 53)}{(x + 2) (x - 15)}$

Step 6.3

Factor $- 1$ out of $- (3 x) - 1 (- 53)$ .

$\frac{- (3 x - 53)}{(x + 2) (x - 15)}$

Step 6.4

Simplify the expression.

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Step 6.4.1

Rewrite $- (3 x - 53)$ as $- 1 (3 x - 53)$ .

$\frac{- 1 (3 x - 53)}{(x + 2) (x - 15)}$

Step 6.4.2

Move the negative in front of the fraction.

$- \frac{3 x - 53}{(x + 2) (x - 15)}$