Solve 4/x^2-7x+1/x^2-4x-21

$$\frac { 4 } { x ^ { 2 } - 7 x } + \frac { 1 } { x ^ { 2 } - 4 x - 21 }$$

Evaluate

$$\frac{5x+12}{x\left(x-7\right)\left(x+3\right)}$$

Short Solution Steps

Factor x^{2}-7x. Factor x^{2}-4x-21.

$$\frac{4}{x\left(x-7\right)}+\frac{1}{\left(x-7\right)\left(x+3\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x-7\right) and \left(x-7\right)\left(x+3\right) is x\left(x-7\right)\left(x+3\right). Multiply \frac{4}{x\left(x-7\right)} times \frac{x+3}{x+3}. Multiply \frac{1}{\left(x-7\right)\left(x+3\right)} times \frac{x}{x}.

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

Since \frac{4\left(x+3\right)}{x\left(x-7\right)\left(x+3\right)} and \frac{x}{x\left(x-7\right)\left(x+3\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 4\left(x+3\right)+x.

$$\frac{4x+12+x}{x\left(x-7\right)\left(x+3\right)}$$

Combine like terms in 4x+12+x.

$$\frac{5x+12}{x\left(x-7\right)\left(x+3\right)}$$

Expand x\left(x-7\right)\left(x+3\right).

$$\frac{5x+12}{x^{3}-4x^{2}-21x}$$

Differentiate w.r.t. x

$$\frac{2\left(126+48x-8x^{2}-5x^{3}\right)}{\left(x\left(x-7\right)\left(x+3\right)\right)^{2}}$$