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

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

Evaluate

$$\frac{24-3x-2x^{2}}{\left(x-2\right)\left(x+3\right)}$$

Short Solution Steps

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

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

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

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

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

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

Combine like terms in 2x+6-3x+6.

$$\frac{-x+12}{\left(x-2\right)\left(x+3\right)}-2$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 2 times \frac{\left(x-2\right)\left(x+3\right)}{\left(x-2\right)\left(x+3\right)}.

$$\frac{-x+12}{\left(x-2\right)\left(x+3\right)}-\frac{2\left(x-2\right)\left(x+3\right)}{\left(x-2\right)\left(x+3\right)}$$

Since \frac{-x+12}{\left(x-2\right)\left(x+3\right)} and \frac{2\left(x-2\right)\left(x+3\right)}{\left(x-2\right)\left(x+3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-x+12-2\left(x-2\right)\left(x+3\right)}{\left(x-2\right)\left(x+3\right)}$$

Do the multiplications in -x+12-2\left(x-2\right)\left(x+3\right).

$$\frac{-x+12-2x^{2}-6x+4x+12}{\left(x-2\right)\left(x+3\right)}$$

Combine like terms in -x+12-2x^{2}-6x+4x+12.

$$\frac{-3x+24-2x^{2}}{\left(x-2\right)\left(x+3\right)}$$

Expand \left(x-2\right)\left(x+3\right).

$$\frac{-3x+24-2x^{2}}{x^{2}+x-6}$$

Differentiate w.r.t. x

$$\frac{x^{2}-24x-6}{x^{4}+2x^{3}-11x^{2}-12x+36}$$