Precalculus Examples

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

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

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Factor

x^{2} + 3 x + 2

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

Tap for more steps...

Step 1.1.1

Consider the form

x^{2} + b x + c

$x^{2} + b x + c$ . Find a pair of integers whose product is

c

$c$ and whose sum is

b

$b$ . In this case, whose product is

2

$2$ and whose sum is

3

$3$ .

1, 2

$1, 2$

Step 1.1.2

Write the factored form using these integers.

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

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

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

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

Step 1.2

Factor

x^{2} - 2 x - 3

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

Tap for more steps...

Step 1.2.1

Consider the form

x^{2} + b x + c

$x^{2} + b x + c$ . Find a pair of integers whose product is

c

$c$ and whose sum is

b

$b$ . In this case, whose product is

- 3

$- 3$ and whose sum is

- 2

$- 2$ .

- 3, 1

$- 3, 1$

Step 1.2.2

Write the factored form using these integers.

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

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

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

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

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

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

Step 2

To write

\frac{1}{(x + 1) (x + 2)}

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

\frac{x - 3}{x - 3}

$\frac{x - 3}{x - 3}$ .

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

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

Step 3

To write

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

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

\frac{x + 2}{x + 2}

$\frac{x + 2}{x + 2}$ .

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

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

Step 4

Write each expression with a common denominator of

(x + 1) (x + 2) (x - 3)

$(x + 1) (x + 2) (x - 3)$ , by multiplying each by an appropriate factor of

1

$1$ .

Tap for more steps...

Step 4.1

Multiply

\frac{1}{(x + 1) (x + 2)}

$\frac{1}{(x + 1) (x + 2)}$ by

\frac{x - 3}{x - 3}

$\frac{x - 3}{x - 3}$ .

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

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

Step 4.2

Multiply

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

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

\frac{x + 2}{x + 2}

$\frac{x + 2}{x + 2}$ .

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

$(x - 3) (x + 1) (x + 2)$ .

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

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

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

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

Step 5

Combine the numerators over the common denominator.

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

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

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)}

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

Step 6.2

Multiply

- 1

$- 1$ by

2

$2$ .

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

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

Step 6.3

Subtract

x

$x$ from

x

$x$ .

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

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

Step 6.4

Subtract

3

$3$ from

0

$0$ .

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

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

Step 6.5

Subtract

2

$2$ from

- 3

$- 3$ .

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

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

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

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