Simplify 3/(x^2-x-2)-2/(x^2+x-6)

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

Step 1

Simplify each term.

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

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 $- 1$ .

$- 2, 1$

Write the factored form using these integers.

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

Factor $x^{2} + x - 6$ using the AC method.

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 $1$ .

$- 2, 3$

Write the factored form using these integers.

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

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

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

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

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

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

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

Apply the distributive property.

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

Multiply $3$ by $3$ .

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

Apply the distributive property.

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

Multiply $- 2$ by $1$ .

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

Subtract $2 x$ from $3 x$ .

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

Subtract $2$ from $9$ .

$\frac{x + 7}{(x + 1) (x + 3) (x - 2)}$

Simplify 3/(x^2-x-2)-2/(x^2+x-6)

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