Simplify (x+2)/(x^2+4x+3)-(5x)/(x^2-9)

$\frac{x + 2}{x^{2} + 4 x + 3} - \frac{5 x}{x^{2} - 9}$

Simplify each term.

Factor $x^{2} + 4 x + 3$ 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 $4$ .

$1, 3$

Write the factored form using these integers.

$\frac{x + 2}{(x + 1) (x + 3)} - \frac{5 x}{x^{2} - 9}$

Simplify the denominator.

Rewrite $9$ as $3^{2}$ .

$\frac{x + 2}{(x + 1) (x + 3)} - \frac{5 x}{x^{2} - 3^{2}}$

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 $b = 3$ .

$\frac{x + 2}{(x + 1) (x + 3)} - \frac{5 x}{(x + 3) (x - 3)}$

To write $\frac{x + 2}{(x + 1) (x + 3)}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

$\frac{x + 2}{(x + 1) (x + 3)} \cdot \frac{x - 3}{x - 3} - \frac{5 x}{(x + 3) (x - 3)}$

To write $- \frac{5 x}{(x + 3) (x - 3)}$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

$\frac{x + 2}{(x + 1) (x + 3)} \cdot \frac{x - 3}{x - 3} - \frac{5 x}{(x + 3) (x - 3)} \cdot \frac{x + 1}{x + 1}$

Write each expression with a common denominator of $(x + 1) (x + 3) (x - 3)$ , by multiplying each by an appropriate factor of $1$ .

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

$\frac{(x + 2) (x - 3)}{(x + 1) (x + 3) (x - 3)} - \frac{5 x}{(x + 3) (x - 3)} \cdot \frac{x + 1}{x + 1}$

Multiply $\frac{5 x}{(x + 3) (x - 3)}$ by $\frac{x + 1}{x + 1}$ .

$\frac{(x + 2) (x - 3)}{(x + 1) (x + 3) (x - 3)} - \frac{5 x (x + 1)}{(x + 3) (x - 3) (x + 1)}$

Reorder the factors of $(x + 3) (x - 3) (x + 1)$ .

$\frac{(x + 2) (x - 3)}{(x + 1) (x + 3) (x - 3)} - \frac{5 x (x + 1)}{(x + 1) (x + 3) (x - 3)}$

Combine the numerators over the common denominator.

$\frac{(x + 2) (x - 3) - 5 x (x + 1)}{(x + 1) (x + 3) (x - 3)}$

Simplify the numerator.

Expand $(x + 2) (x - 3)$ using the FOIL Method.

Apply the distributive property.

$\frac{x (x - 3) + 2 (x - 3) - 5 x (x + 1)}{(x + 1) (x + 3) (x - 3)}$

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 3 + 2 (x - 3) - 5 x (x + 1)}{(x + 1) (x + 3) (x - 3)}$

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 3 + 2 x + 2 \cdot - 3 - 5 x (x + 1)}{(x + 1) (x + 3) (x - 3)}$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot - 3 + 2 x + 2 \cdot - 3 - 5 x (x + 1)}{(x + 1) (x + 3) (x - 3)}$

Move $- 3$ to the left of $x$ .

$\frac{x^{2} - 3 \cdot x + 2 x + 2 \cdot - 3 - 5 x (x + 1)}{(x + 1) (x + 3) (x - 3)}$

Multiply $2$ by $- 3$ .

$\frac{x^{2} - 3 x + 2 x - 6 - 5 x (x + 1)}{(x + 1) (x + 3) (x - 3)}$

Add $- 3 x$ and $2 x$ .

$\frac{x^{2} - x - 6 - 5 x (x + 1)}{(x + 1) (x + 3) (x - 3)}$

Apply the distributive property.

$\frac{x^{2} - x - 6 - 5 x \cdot x - 5 x \cdot 1}{(x + 1) (x + 3) (x - 3)}$

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

Move $x$ .

$\frac{x^{2} - x - 6 - 5 (x \cdot x) - 5 x \cdot 1}{(x + 1) (x + 3) (x - 3)}$

Multiply $x$ by $x$ .

$\frac{x^{2} - x - 6 - 5 x^{2} - 5 x \cdot 1}{(x + 1) (x + 3) (x - 3)}$

Multiply $- 5$ by $1$ .

$\frac{x^{2} - x - 6 - 5 x^{2} - 5 x}{(x + 1) (x + 3) (x - 3)}$

Subtract $5 x^{2}$ from $x^{2}$ .

$\frac{- 4 x^{2} - x - 6 - 5 x}{(x + 1) (x + 3) (x - 3)}$

Subtract $5 x$ from $- x$ .

$\frac{- 4 x^{2} - 6 x - 6}{(x + 1) (x + 3) (x - 3)}$

Factor $2$ out of $- 4 x^{2} - 6 x - 6$ .

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

$\frac{2 (- 2 x^{2}) - 6 x - 6}{(x + 1) (x + 3) (x - 3)}$

Factor $2$ out of $- 6 x$ .

$\frac{2 (- 2 x^{2}) + 2 (- 3 x) - 6}{(x + 1) (x + 3) (x - 3)}$

Factor $2$ out of $- 6$ .

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

Factor $2$ out of $2 (- 2 x^{2}) + 2 (- 3 x)$ .

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

Factor $2$ out of $2 (- 2 x^{2} - 3 x) + 2 (- 3)$ .

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

Simplify with factoring out.

Factor $- 1$ out of $- 2 x^{2}$ .

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

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

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

Factor $- 1$ out of $- (2 x^{2}) - (3 x)$ .

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

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

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

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

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

Simplify the expression.

Rewrite $- (2 x^{2} + 3 x + 3)$ as $- 1 (2 x^{2} + 3 x + 3)$ .

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

Move the negative in front of the fraction.

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

Simplify (x+2)/(x^2+4x+3)-(5x)/(x^2-9)

Related Questions

Support LCMGCF.com by disabling your adblocker.