Simplify (2x)/(x^2-x-2)-(4x)/(x^2-3x+2)

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

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.

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

Factor $x^{2} - 3 x + 2$ 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 $- 3$ .

$- 2, - 1$

Write the factored form using these integers.

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

Step 2

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

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

Step 3

To write $- \frac{4 x}{(x - 2) (x - 1)}$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

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

Step 4

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

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

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

Multiply $\frac{4 x}{(x - 2) (x - 1)}$ by $\frac{x + 1}{x + 1}$ .

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

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

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

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

Factor $2 x$ out of $2 x (x - 1) - 4 x (x + 1)$ .

Factor $2 x$ out of $- 4 x (x + 1)$ .

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

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

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

Apply the distributive property.

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

Multiply $- 2$ by $1$ .

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

Subtract $2 x$ from $x$ .

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

Subtract $2$ from $- 1$ .

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

Step 7

Simplify with factoring out.

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

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

Rewrite $- 3$ as $- 1 (3)$ .

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

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

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

Simplify the expression.

Rewrite $- (x + 3)$ as $- 1 (x + 3)$ .

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

Move the negative in front of the fraction.

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

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

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

Simplify (2x)/(x^2-x-2)-(4x)/(x^2-3x+2)

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