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

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

Step 1

Factor $x^{2} - 2 x - 3$ 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 $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Write the factored form using these integers.

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

Step 2

To write $- \frac{6 x}{x - 3}$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

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

Step 3

Simplify terms.

Multiply $\frac{6 x}{x - 3}$ by $\frac{x + 1}{x + 1}$ .

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

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

Factor $3 x$ out of $3 x - 6 x (x + 1)$ .

Factor $3 x$ out of $3 x$ .

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

Factor $3 x$ out of $- 6 x (x + 1)$ .

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

Factor $3 x$ out of $3 x (1) + 3 x (- 2 (x + 1))$ .

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

Apply the distributive property.

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

Multiply $- 2$ by $1$ .

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

Subtract $2$ from $1$ .

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

Step 5

Simplify with factoring out.

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

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

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

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

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

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

Simplify the expression.

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

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

Move the negative in front of the fraction.

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

Reorder factors in $- \frac{(3 x) (2 x + 1)}{(x - 3) (x + 1)}$ .

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

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

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