Simplify (x+2)/(x^2+4x)+4/(x^2-16)

$\frac{x + 2}{x^{2} + 4 x} + \frac{4}{x^{2} - 16}$

Step 1

Simplify each term.

Factor $x$ out of $x^{2} + 4 x$ .

Factor $x$ out of $x^{2}$ .

$\frac{x + 2}{x \cdot x + 4 x} + \frac{4}{x^{2} - 16}$

Factor $x$ out of $4 x$ .

$\frac{x + 2}{x \cdot x + x \cdot 4} + \frac{4}{x^{2} - 16}$

Factor $x$ out of $x \cdot x + x \cdot 4$ .

$\frac{x + 2}{x (x + 4)} + \frac{4}{x^{2} - 16}$

Simplify the denominator.

Rewrite $16$ as $4^{2}$ .

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

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 $b = 4$ .

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

Step 2

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

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

Step 3

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

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

Step 4

Write each expression with a common denominator of $x (x + 4) (x - 4)$ , by multiplying each by an appropriate factor of $1$ .

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

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

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

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

Reorder the factors of $(x + 4) (x - 4) x$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

Expand $(x + 2) (x - 4)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $- 4$ to the left of $x$ .

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

Multiply $2$ by $- 4$ .

$\frac{x^{2} - 4 x + 2 x - 8 + 4 x}{x (x + 4) (x - 4)}$

Add $- 4 x$ and $2 x$ .

$\frac{x^{2} - 2 x - 8 + 4 x}{x (x + 4) (x - 4)}$

Add $- 2 x$ and $4 x$ .

$\frac{x^{2} + 2 x - 8}{x (x + 4) (x - 4)}$

Factor $x^{2} + 2 x - 8$ 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 $- 8$ and whose sum is $2$ .

$- 2, 4$

Write the factored form using these integers.

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

Step 7

Cancel the common factor of $x + 4$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify (x+2)/(x^2+4x)+4/(x^2-16)

Related Questions

Support LCMGCF.com by disabling your adblocker.