Simplify (x^2)/(x^2-4)-(x+1)/(x+2)

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

Step 1

Simplify the denominator.

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

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

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 $b = 2$ .

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

Step 2

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

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

Step 3

Simplify terms.

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

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

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

Apply the distributive property.

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

Multiply $- 1$ by $1$ .

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

Expand $(- x - 1) (x - 2)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 2$ by $- 1$ .

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

Rewrite $- 1 x$ as $- x$ .

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

Multiply $- 1$ by $- 2$ .

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

Subtract $x$ from $2 x$ .

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

Subtract $x^{2}$ from $x^{2}$ .

$\frac{0 + x + 2}{(x + 2) (x - 2)}$

Add $0$ and $x$ .

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

Step 5

Cancel the common factor of $x + 2$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{1}{x - 2}$

Simplify (x^2)/(x^2-4)-(x+1)/(x+2)

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