Simplify (5x)/(x+8)+(4x-9)/(x^2+5x-24)

$\frac{5 x}{x + 8} + \frac{4 x - 9}{x^{2} + 5 x - 24}$

Factor $x^{2} + 5 x - 24$ 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 $- 24$ and whose sum is $5$ .

$- 3, 8$

Write the factored form using these integers.

$\frac{5 x}{x + 8} + \frac{4 x - 9}{(x - 3) (x + 8)}$

To write $\frac{5 x}{x + 8}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

$\frac{5 x}{x + 8} \cdot \frac{x - 3}{x - 3} + \frac{4 x - 9}{(x - 3) (x + 8)}$

Write each expression with a common denominator of $(x + 8) (x - 3)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{5 x}{x + 8}$ by $\frac{x - 3}{x - 3}$ .

$\frac{5 x (x - 3)}{(x + 8) (x - 3)} + \frac{4 x - 9}{(x - 3) (x + 8)}$

Reorder the factors of $(x - 3) (x + 8)$ .

$\frac{5 x (x - 3)}{(x + 8) (x - 3)} + \frac{4 x - 9}{(x + 8) (x - 3)}$

Combine the numerators over the common denominator.

$\frac{5 x (x - 3) + 4 x - 9}{(x + 8) (x - 3)}$

Simplify the numerator.

Apply the distributive property.

$\frac{5 x \cdot x + 5 x \cdot - 3 + 4 x - 9}{(x + 8) (x - 3)}$

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

Move $x$ .

$\frac{5 (x \cdot x) + 5 x \cdot - 3 + 4 x - 9}{(x + 8) (x - 3)}$

Multiply $x$ by $x$ .

$\frac{5 x^{2} + 5 x \cdot - 3 + 4 x - 9}{(x + 8) (x - 3)}$

Multiply $- 3$ by $5$ .

$\frac{5 x^{2} - 15 x + 4 x - 9}{(x + 8) (x - 3)}$

Add $- 15 x$ and $4 x$ .

$\frac{5 x^{2} - 11 x - 9}{(x + 8) (x - 3)}$

Simplify (5x)/(x+8)+(4x-9)/(x^2+5x-24)

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