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

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

Step 1

Simplify each term.

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 $3$ .

$1, 2$

Write the factored form using these integers.

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

Simplify the denominator.

Rewrite $4$ as $2^{2}$ .

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

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 $b = 2$ .

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

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

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

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

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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

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

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

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

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

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

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

Apply the distributive property.

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

Multiply $2$ by $- 2$ .

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

Apply the distributive property.

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

Multiply $- 1$ by $1$ .

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

Subtract $x$ from $2 x$ .

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

Subtract $1$ from $- 4$ .

$\frac{x (x - 5)}{(x + 1) (x + 2) (x - 2)}$

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

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