Precalculus Examples

$\frac{1}{x^{2} + 3 x + 2} - \frac{1}{x^{2} - 2 x - 3}$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Factor $x^{2} + 3 x + 2$ using the AC method.

Tap for more steps...

Step 1.1.1

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 $2$ and whose sum is $3$ .

$1, 2$

Step 1.1.2

Write the factored form using these integers.

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

Step 1.2

Factor $x^{2} - 2 x - 3$ using the AC method.

Tap for more steps...

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 $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Step 1.2.2

Write the factored form using these integers.

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

Step 2

To write $\frac{1}{(x + 1) (x + 2)}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

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

Step 3

To write $- \frac{1}{(x - 3) (x + 1)}$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

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

Step 4

Write each expression with a common denominator of $(x + 1) (x + 2) (x - 3)$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 4.1

Multiply $\frac{1}{(x + 1) (x + 2)}$ by $\frac{x - 3}{x - 3}$ .

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

Step 4.2

Multiply $\frac{1}{(x - 3) (x + 1)}$ by $\frac{x + 2}{x + 2}$ .

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

Step 4.3

Reorder the factors of $(x - 3) (x + 1) (x + 2)$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

Tap for more steps...

Step 6.1

Apply the distributive property.

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

Step 6.2

Multiply $- 1$ by $2$ .

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

Step 6.3

Subtract $x$ from $x$ .

$\frac{0 - 3 - 2}{(x + 1) (x + 2) (x - 3)}$

Step 6.4

Subtract $3$ from $0$ .

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

Step 6.5

Subtract $2$ from $- 3$ .

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

Step 7

Move the negative in front of the fraction.

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