Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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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 $1$ .

$- 1, 2$

Step 1.1.2

Write the factored form using these integers.

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

Step 1.2

Factor $x^{2} - x - 6$ 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 $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Step 1.2.2

Write the factored form using these integers.

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

Step 2

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

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

Step 3

To write $- \frac{5}{(x - 3) (x + 2)}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Multiply $3$ by $- 3$ .

$\frac{3 x - 9 - 5 (x - 1)}{(x + 2) (x - 1) (x - 3)}$

Step 6.3

Apply the distributive property.

$\frac{3 x - 9 - 5 x - 5 \cdot - 1}{(x + 2) (x - 1) (x - 3)}$

Step 6.4

Multiply $- 5$ by $- 1$ .

$\frac{3 x - 9 - 5 x + 5}{(x + 2) (x - 1) (x - 3)}$

Step 6.5

Subtract $5 x$ from $3 x$ .

$\frac{- 2 x - 9 + 5}{(x + 2) (x - 1) (x - 3)}$

Step 6.6

Add $- 9$ and $5$ .

$\frac{- 2 x - 4}{(x + 2) (x - 1) (x - 3)}$

Step 6.7

Factor $2$ out of $- 2 x - 4$ .

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Step 6.7.1

Factor $2$ out of $- 2 x$ .

$\frac{2 (- x) - 4}{(x + 2) (x - 1) (x - 3)}$

Step 6.7.2

Factor $2$ out of $- 4$ .

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

Step 6.7.3

Factor $2$ out of $2 (- x) + 2 (- 2)$ .

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

Step 7

Cancel the common factor of $- x - 2$ and $x + 2$ .

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

Factor $- 1$ out of $- (x) - 1 (2)$ .

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

Step 7.4

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

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

Step 7.5

Cancel the common factor.

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

Step 7.6

Rewrite the expression.

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

Step 8

Multiply $2$ by $- 1$ .

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

Step 9

Move the negative in front of the fraction.

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