Solve 2/x^2+7x+10+3/x^2-25

$$\frac { 2 } { x ^ { 2 } + 7 x + 10 } + \frac { 3 } { x ^ { 2 } - 25 }$$

Evaluate

$$\frac{5x-4}{\left(x+2\right)\left(x^{2}-25\right)}$$

Short Solution Steps

Factor x^{2}+7x+10. Factor x^{2}-25.

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

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

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

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

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

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

$$\frac{2x-10+3x+6}{\left(x-5\right)\left(x+2\right)\left(x+5\right)}$$

Combine like terms in 2x-10+3x+6.

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

Expand \left(x-5\right)\left(x+2\right)\left(x+5\right).

$$\frac{5x-4}{x^{3}+2x^{2}-25x-50}$$

Differentiate w.r.t. x

$$\frac{2\left(-5x^{3}+x^{2}+8x-175\right)}{\left(\left(x+2\right)\left(x^{2}-25\right)\right)^{2}}$$