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

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

Simplify each term.

Simplify the denominator.

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

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

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 $b = 2$ .

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

Factor $x^{2} + x - 6$ 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 $1$ .

$- 2, 3$

Write the factored form using these integers.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

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

Factor $5$ out of $5 x (x + 3)$ .

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

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

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

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

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

Apply the distributive property.

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

Multiply $x$ by $x$ .

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

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

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

Apply the distributive property.

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

Multiply $- 1$ by $2$ .

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

Subtract $x$ from $3 x$ .

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

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

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