Simplify 5/(x^2+x-6)-2/(x-2)

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

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.

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

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

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

Simplify terms.

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $- 2$ by $3$ .

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

Subtract $6$ from $5$ .

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

Simplify with factoring out.

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

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

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

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

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

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

Simplify the expression.

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

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

Move the negative in front of the fraction.

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

Simplify 5/(x^2+x-6)-2/(x-2)

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