Solve 5/x^2-9x+8+3/x^2-6x-16

$$\frac{ 5 }{ { x }^{ 2 } -9x+8 } + \frac{ 3 }{ { x }^{ 2 } -6x-16 }$$

Evaluate

$$\frac{8x+7}{\left(x-8\right)\left(x-1\right)\left(x+2\right)}$$

Short Solution Steps

Factor x^{2}-9x+8. Factor x^{2}-6x-16.

$$\frac{5}{\left(x-8\right)\left(x-1\right)}+\frac{3}{\left(x-8\right)\left(x+2\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-8\right)\left(x-1\right) and \left(x-8\right)\left(x+2\right) is \left(x-8\right)\left(x-1\right)\left(x+2\right). Multiply \frac{5}{\left(x-8\right)\left(x-1\right)} times \frac{x+2}{x+2}. Multiply \frac{3}{\left(x-8\right)\left(x+2\right)} times \frac{x-1}{x-1}.

$$\frac{5\left(x+2\right)}{\left(x-8\right)\left(x-1\right)\left(x+2\right)}+\frac{3\left(x-1\right)}{\left(x-8\right)\left(x-1\right)\left(x+2\right)}$$

Since \frac{5\left(x+2\right)}{\left(x-8\right)\left(x-1\right)\left(x+2\right)} and \frac{3\left(x-1\right)}{\left(x-8\right)\left(x-1\right)\left(x+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{5\left(x+2\right)+3\left(x-1\right)}{\left(x-8\right)\left(x-1\right)\left(x+2\right)}$$

Do the multiplications in 5\left(x+2\right)+3\left(x-1\right).

$$\frac{5x+10+3x-3}{\left(x-8\right)\left(x-1\right)\left(x+2\right)}$$

Combine like terms in 5x+10+3x-3.

$$\frac{8x+7}{\left(x-8\right)\left(x-1\right)\left(x+2\right)}$$

Expand \left(x-8\right)\left(x-1\right)\left(x+2\right).

$$\frac{8x+7}{x^{3}-7x^{2}-10x+16}$$

Differentiate w.r.t. x

$$\frac{198+98x+35x^{2}-16x^{3}}{\left(\left(x-8\right)\left(x-1\right)\left(x+2\right)\right)^{2}}$$