Solve x/x-2-1/x+6-1/x^2+4x-12

$$\frac { x } { x - 2 } - \frac { 1 } { x + 6 } - \frac { 1 } { x ^ { 2 } + 4 x - 12 }$$

Evaluate

$$\frac{x^{2}+5x+1}{\left(x-2\right)\left(x+6\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-2 and x+6 is \left(x-2\right)\left(x+6\right). Multiply \frac{x}{x-2} times \frac{x+6}{x+6}. Multiply \frac{1}{x+6} times \frac{x-2}{x-2}.

$$\frac{x\left(x+6\right)}{\left(x-2\right)\left(x+6\right)}-\frac{x-2}{\left(x-2\right)\left(x+6\right)}-\frac{1}{x^{2}+4x-12}$$

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

$$\frac{x\left(x+6\right)-\left(x-2\right)}{\left(x-2\right)\left(x+6\right)}-\frac{1}{x^{2}+4x-12}$$

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

$$\frac{x^{2}+6x-x+2}{\left(x-2\right)\left(x+6\right)}-\frac{1}{x^{2}+4x-12}$$

Combine like terms in x^{2}+6x-x+2.

$$\frac{x^{2}+5x+2}{\left(x-2\right)\left(x+6\right)}-\frac{1}{x^{2}+4x-12}$$

Factor x^{2}+4x-12.

$$\frac{x^{2}+5x+2}{\left(x-2\right)\left(x+6\right)}-\frac{1}{\left(x-2\right)\left(x+6\right)}$$

Since \frac{x^{2}+5x+2}{\left(x-2\right)\left(x+6\right)} and \frac{1}{\left(x-2\right)\left(x+6\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x^{2}+5x+2-1}{\left(x-2\right)\left(x+6\right)}$$

Combine like terms in x^{2}+5x+2-1.

$$\frac{x^{2}+5x+1}{\left(x-2\right)\left(x+6\right)}$$

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

$$\frac{x^{2}+5x+1}{x^{2}+4x-12}$$

Differentiate w.r.t. x

$$-\frac{\left(x+13\right)^{2}-105}{\left(\left(x-2\right)\left(x+6\right)\right)^{2}}$$