Simplify (5x)/(x^2-9)-(4x)/(x^2+5x+6)

$\frac{5 x}{x^{2} - 9} - \frac{4 x}{x^{2} + 5 x + 6}$

Step 1

Simplify each term.

Simplify the denominator.

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

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

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 $b = 3$ .

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

Factor $x^{2} + 5 x + 6$ 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 $6$ and whose sum is $5$ .

$2, 3$

Write the factored form using these integers.

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

Step 2

To write $\frac{5 x}{(x + 3) (x - 3)}$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

$\frac{5 x}{(x + 3) (x - 3)} \cdot \frac{x + 2}{x + 2} - \frac{4 x}{(x + 2) (x + 3)}$

Step 3

To write $- \frac{4 x}{(x + 2) (x + 3)}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

$\frac{5 x}{(x + 3) (x - 3)} \cdot \frac{x + 2}{x + 2} - \frac{4 x}{(x + 2) (x + 3)} \cdot \frac{x - 3}{x - 3}$

Step 4

Write each expression with a common denominator of $(x + 3) (x - 3) (x + 2)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{5 x}{(x + 3) (x - 3)}$ by $\frac{x + 2}{x + 2}$ .

$\frac{5 x (x + 2)}{(x + 3) (x - 3) (x + 2)} - \frac{4 x}{(x + 2) (x + 3)} \cdot \frac{x - 3}{x - 3}$

Multiply $\frac{4 x}{(x + 2) (x + 3)}$ by $\frac{x - 3}{x - 3}$ .

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

Reorder the factors of $(x + 3) (x - 3) (x + 2)$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

Factor $x$ out of $5 x (x + 2) - 4 x (x - 3)$ .

Factor $x$ out of $5 x (x + 2)$ .

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

Factor $x$ out of $- 4 x (x - 3)$ .

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

Factor $x$ out of $x (5 (x + 2)) + x (- 4 (x - 3))$ .

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

Apply the distributive property.

$\frac{x (5 x + 5 \cdot 2 - 4 (x - 3))}{(x + 2) (x + 3) (x - 3)}$

Multiply $5$ by $2$ .

$\frac{x (5 x + 10 - 4 (x - 3))}{(x + 2) (x + 3) (x - 3)}$

Apply the distributive property.

$\frac{x (5 x + 10 - 4 x - 4 \cdot - 3)}{(x + 2) (x + 3) (x - 3)}$

Multiply $- 4$ by $- 3$ .

$\frac{x (5 x + 10 - 4 x + 12)}{(x + 2) (x + 3) (x - 3)}$

Subtract $4 x$ from $5 x$ .

$\frac{x (x + 10 + 12)}{(x + 2) (x + 3) (x - 3)}$

Add $10$ and $12$ .

$\frac{x (x + 22)}{(x + 2) (x + 3) (x - 3)}$

Simplify (5x)/(x^2-9)-(4x)/(x^2+5x+6)

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