Solve 1/x^2-10x+16+3/x^2+14x-32

$$\frac { 1 } { x ^ { 2 } - 10 x + 16 } + \frac { 3 } { x ^ { 2 } + 14 x - 32 }$$

Evaluate

$$\frac{4}{\left(x-8\right)\left(x+16\right)}$$

Short Solution Steps

Factor x^{2}-10x+16. Factor x^{2}+14x-32.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-8\right)\left(x-2\right) and \left(x-2\right)\left(x+16\right) is \left(x-8\right)\left(x-2\right)\left(x+16\right). Multiply \frac{1}{\left(x-8\right)\left(x-2\right)} times \frac{x+16}{x+16}. Multiply \frac{3}{\left(x-2\right)\left(x+16\right)} times \frac{x-8}{x-8}.

$$\frac{x+16}{\left(x-8\right)\left(x-2\right)\left(x+16\right)}+\frac{3\left(x-8\right)}{\left(x-8\right)\left(x-2\right)\left(x+16\right)}$$

Since \frac{x+16}{\left(x-8\right)\left(x-2\right)\left(x+16\right)} and \frac{3\left(x-8\right)}{\left(x-8\right)\left(x-2\right)\left(x+16\right)} have the same denominator, add them by adding their numerators.

$$\frac{x+16+3\left(x-8\right)}{\left(x-8\right)\left(x-2\right)\left(x+16\right)}$$

Do the multiplications in x+16+3\left(x-8\right).

$$\frac{x+16+3x-24}{\left(x-8\right)\left(x-2\right)\left(x+16\right)}$$

Combine like terms in x+16+3x-24.

$$\frac{4x-8}{\left(x-8\right)\left(x-2\right)\left(x+16\right)}$$

Factor the expressions that are not already factored in \frac{4x-8}{\left(x-8\right)\left(x-2\right)\left(x+16\right)}.

$$\frac{4\left(x-2\right)}{\left(x-8\right)\left(x-2\right)\left(x+16\right)}$$

Cancel out x-2 in both numerator and denominator.

$$\frac{4}{\left(x-8\right)\left(x+16\right)}$$

Expand \left(x-8\right)\left(x+16\right).

$$\frac{4}{x^{2}+8x-128}$$

Differentiate w.r.t. x

$$\frac{8\left(-x-4\right)}{\left(\left(x-8\right)\left(x+16\right)\right)^{2}}$$