Question

Find each sum or difference.

$\frac{5}{x^2-2x-24}+\frac{x}{x-6}$

Short answer

$\frac{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathbf{4}\mathbf{x}\mathbf{+}\mathbf{5}}{\left(x+4\right)\left(x-6\right)}$

Step-by-step solution

Step 1. Find LCD.

As$x^2-2x-24=\left(x+4\right)\left(x-6\right)$the LCD$\left(x+4\right)\left(x-6\right)$

Step 2. Factor.

Factor $x^2-2x-24$.

$\frac{5}{x^2-2x-24}+\frac{x}{x-6}=\frac{5}{\left(x+4\right)\left(x-6\right)}+\frac{x}{x-6}$

Step 3. Write using LCD.

Write $\frac{x}{x-6}$using LCD.

$\frac{5}{\left(x+4\right)\left(x-6\right)}+\frac{x\left(x+4\right)}{\left(x-6\right)\left(x+4\right)}$

Step 4. Simplify.

Simplify the above-obtained expression.

$\frac{5+x\left(x+4\right)}{\left(x+4\right)\left(x-6\right)}=\frac{x^2+4x+5}{\left(x+4\right)\left(x-6\right)}$

Therefore, the sum of the given expression is $\frac{x^2+4x+5}{\left(x+4\right)\left(x-6\right)}$.