Solve 7/x^2-5x+6-2/x^2-3x

$$\frac { 7 } { x ^ { 2 } - 5 x + 6 } - \frac { 2 } { x ^ { 2 } - 3 x }$$

Evaluate

$$\frac{5x+4}{x\left(x-3\right)\left(x-2\right)}$$

Short Solution Steps

Factor x^{2}-5x+6. Factor x^{2}-3x.

$$\frac{7}{\left(x-3\right)\left(x-2\right)}-\frac{2}{x\left(x-3\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-3\right)\left(x-2\right) and x\left(x-3\right) is x\left(x-3\right)\left(x-2\right). Multiply \frac{7}{\left(x-3\right)\left(x-2\right)} times \frac{x}{x}. Multiply \frac{2}{x\left(x-3\right)} times \frac{x-2}{x-2}.

$$\frac{7x}{x\left(x-3\right)\left(x-2\right)}-\frac{2\left(x-2\right)}{x\left(x-3\right)\left(x-2\right)}$$

Since \frac{7x}{x\left(x-3\right)\left(x-2\right)} and \frac{2\left(x-2\right)}{x\left(x-3\right)\left(x-2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{7x-2\left(x-2\right)}{x\left(x-3\right)\left(x-2\right)}$$

Do the multiplications in 7x-2\left(x-2\right).

$$\frac{7x-2x+4}{x\left(x-3\right)\left(x-2\right)}$$

Combine like terms in 7x-2x+4.

$$\frac{5x+4}{x\left(x-3\right)\left(x-2\right)}$$

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

$$\frac{5x+4}{x^{3}-5x^{2}+6x}$$

Differentiate w.r.t. x

$$\frac{-10x^{3}+13x^{2}+40x-24}{\left(x\left(x-3\right)\left(x-2\right)\right)^{2}}$$