Algebra Examples

$\frac{x + 2}{x^{2} - 3 x - 18} - \frac{5 x}{x - 6}$

Step 1

Factor $x^{2} - 3 x - 18$ 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 $- 18$ and whose sum is $- 3$ .

$- 6, 3$

Step 1.2

Write the factored form using these integers.

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

Step 2

To write $- \frac{5 x}{x - 6}$ as a fraction with a common denominator, multiply by $\frac{x + 3}{x + 3}$ .

$\frac{x + 2}{(x - 6) (x + 3)} - \frac{5 x}{x - 6} \cdot \frac{x + 3}{x + 3}$

Step 3

Simplify terms.

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Step 3.1

Multiply $\frac{5 x}{x - 6}$ by $\frac{x + 3}{x + 3}$ .

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

$\frac{x + 2 - 5 x \cdot x - 5 x \cdot 3}{(x - 6) (x + 3)}$

Step 4.2

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

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Step 4.2.1

Move $x$ .

$\frac{x + 2 - 5 (x \cdot x) - 5 x \cdot 3}{(x - 6) (x + 3)}$

Step 4.2.2

Multiply $x$ by $x$ .

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

Step 4.3

Multiply $3$ by $- 5$ .

$\frac{x + 2 - 5 x^{2} - 15 x}{(x - 6) (x + 3)}$

Step 4.4

Subtract $15 x$ from $x$ .

$\frac{- 14 x + 2 - 5 x^{2}}{(x - 6) (x + 3)}$

Step 4.5

Reorder terms.

$\frac{- 5 x^{2} - 14 x + 2}{(x - 6) (x + 3)}$

Step 5

Simplify with factoring out.

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Step 5.1

Factor $- 1$ out of $- 5 x^{2}$ .

$\frac{- (5 x^{2}) - 14 x + 2}{(x - 6) (x + 3)}$

Step 5.2

Factor $- 1$ out of $- 14 x$ .

$\frac{- (5 x^{2}) - (14 x) + 2}{(x - 6) (x + 3)}$

Step 5.3

Factor $- 1$ out of $- (5 x^{2}) - (14 x)$ .

$\frac{- (5 x^{2} + 14 x) + 2}{(x - 6) (x + 3)}$

Step 5.4

Rewrite $2$ as $- 1 (- 2)$ .

$\frac{- (5 x^{2} + 14 x) - 1 (- 2)}{(x - 6) (x + 3)}$

Step 5.5

Factor $- 1$ out of $- (5 x^{2} + 14 x) - 1 (- 2)$ .

$\frac{- (5 x^{2} + 14 x - 2)}{(x - 6) (x + 3)}$

Step 5.6

Simplify the expression.

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Step 5.6.1

Rewrite $- (5 x^{2} + 14 x - 2)$ as $- 1 (5 x^{2} + 14 x - 2)$ .

$\frac{- 1 (5 x^{2} + 14 x - 2)}{(x - 6) (x + 3)}$

Step 5.6.2

Move the negative in front of the fraction.

$- \frac{5 x^{2} + 14 x - 2}{(x - 6) (x + 3)}$