Solve 9/x^2-9x+20+2/x^2-16

$$\frac { 9 } { x ^ { 2 } - 9 x + 20 } + \frac { 2 } { x ^ { 2 } - 16 }$$

Evaluate

$$\frac{11x+26}{\left(x-5\right)\left(x^{2}-16\right)}$$

Short Solution Steps

Factor x^{2}-9x+20. Factor x^{2}-16.

$$\frac{9}{\left(x-5\right)\left(x-4\right)}+\frac{2}{\left(x-4\right)\left(x+4\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-5\right)\left(x-4\right) and \left(x-4\right)\left(x+4\right) is \left(x-5\right)\left(x-4\right)\left(x+4\right). Multiply \frac{9}{\left(x-5\right)\left(x-4\right)} times \frac{x+4}{x+4}. Multiply \frac{2}{\left(x-4\right)\left(x+4\right)} times \frac{x-5}{x-5}.

$$\frac{9\left(x+4\right)}{\left(x-5\right)\left(x-4\right)\left(x+4\right)}+\frac{2\left(x-5\right)}{\left(x-5\right)\left(x-4\right)\left(x+4\right)}$$

Since \frac{9\left(x+4\right)}{\left(x-5\right)\left(x-4\right)\left(x+4\right)} and \frac{2\left(x-5\right)}{\left(x-5\right)\left(x-4\right)\left(x+4\right)} have the same denominator, add them by adding their numerators.

$$\frac{9\left(x+4\right)+2\left(x-5\right)}{\left(x-5\right)\left(x-4\right)\left(x+4\right)}$$

Do the multiplications in 9\left(x+4\right)+2\left(x-5\right).

$$\frac{9x+36+2x-10}{\left(x-5\right)\left(x-4\right)\left(x+4\right)}$$

Combine like terms in 9x+36+2x-10.

$$\frac{11x+26}{\left(x-5\right)\left(x-4\right)\left(x+4\right)}$$

Expand \left(x-5\right)\left(x-4\right)\left(x+4\right).

$$\frac{11x+26}{x^{3}-5x^{2}-16x+80}$$

Differentiate w.r.t. x

$$\frac{1296+260x-23x^{2}-22x^{3}}{\left(\left(x-5\right)\left(x^{2}-16\right)\right)^{2}}$$