Solve x/x+6-3/x+2

$$\frac { x } { x + 6 } - \frac { 3 } { x + 2 }$$

Evaluate

$$\frac{x^{2}-x-18}{\left(x+2\right)\left(x+6\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+6 and x+2 is \left(x+2\right)\left(x+6\right). Multiply \frac{x}{x+6} times \frac{x+2}{x+2}. Multiply \frac{3}{x+2} times \frac{x+6}{x+6}.

$$\frac{x\left(x+2\right)}{\left(x+2\right)\left(x+6\right)}-\frac{3\left(x+6\right)}{\left(x+2\right)\left(x+6\right)}$$

Since \frac{x\left(x+2\right)}{\left(x+2\right)\left(x+6\right)} and \frac{3\left(x+6\right)}{\left(x+2\right)\left(x+6\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x\left(x+2\right)-3\left(x+6\right)}{\left(x+2\right)\left(x+6\right)}$$

Do the multiplications in x\left(x+2\right)-3\left(x+6\right).

$$\frac{x^{2}+2x-3x-18}{\left(x+2\right)\left(x+6\right)}$$

Combine like terms in x^{2}+2x-3x-18.

$$\frac{x^{2}-x-18}{\left(x+2\right)\left(x+6\right)}$$

Expand \left(x+2\right)\left(x+6\right).

$$\frac{x^{2}-x-18}{x^{2}+8x+12}$$

Differentiate w.r.t. x

$$\frac{3\left(3x^{2}+20x+44\right)}{\left(\left(x+2\right)\left(x+6\right)\right)^{2}}$$