Simplify 2/(x^2-9)-(3x)/(x^2-5x+6)

$\frac{2}{x^{2} - 9} - \frac{3 x}{x^{2} - 5 x + 6}$

Simplify each term.

Simplify the denominator.

Rewrite $9$ as $3^{2}$ .

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

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

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

Factor $x^{2} - 5 x + 6$ 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 $- 5$ .

$- 3, - 2$

Write the factored form using these integers.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $2$ by $- 2$ .

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

Apply the distributive property.

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

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $3$ by $- 3$ .

$\frac{2 x - 4 - 3 x^{2} - 9 x}{(x + 3) (x - 2) (x - 3)}$

Subtract $9 x$ from $2 x$ .

$\frac{- 7 x - 4 - 3 x^{2}}{(x + 3) (x - 2) (x - 3)}$

Reorder terms.

$\frac{- 3 x^{2} - 7 x - 4}{(x + 3) (x - 2) (x - 3)}$

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 3 \cdot - 4 = 12$ and whose sum is $b = - 7$ .

Factor $- 7$ out of $- 7 x$ .

$\frac{- 3 x^{2} - 7 (x) - 4}{(x + 3) (x - 2) (x - 3)}$

Rewrite $- 7$ as $- 3$ plus $- 4$

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

Factor out the greatest common factor (GCF) from each group.

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

Factor the polynomial by factoring out the greatest common factor, $- x - 1$ .

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

Simplify with factoring out.

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

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

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

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

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

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

Simplify the expression.

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

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

Move the negative in front of the fraction.

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

Simplify 2/(x^2-9)-(3x)/(x^2-5x+6)

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