Solve 6/x^2+9x+18+5/x^2+7x+12

$$\frac { 6 } { x ^ { 2 } + 9 x + 18 } + \frac { 5 } { x ^ { 2 } + 7 x + 12 }$$

Evaluate

$$\frac{11x+54}{\left(x+3\right)\left(x+4\right)\left(x+6\right)}$$

Short Solution Steps

Factor x^{2}+9x+18. Factor x^{2}+7x+12.

$$\frac{6}{\left(x+3\right)\left(x+6\right)}+\frac{5}{\left(x+3\right)\left(x+4\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+3\right)\left(x+6\right) and \left(x+3\right)\left(x+4\right) is \left(x+3\right)\left(x+4\right)\left(x+6\right). Multiply \frac{6}{\left(x+3\right)\left(x+6\right)} times \frac{x+4}{x+4}. Multiply \frac{5}{\left(x+3\right)\left(x+4\right)} times \frac{x+6}{x+6}.

$$\frac{6\left(x+4\right)}{\left(x+3\right)\left(x+4\right)\left(x+6\right)}+\frac{5\left(x+6\right)}{\left(x+3\right)\left(x+4\right)\left(x+6\right)}$$

Since \frac{6\left(x+4\right)}{\left(x+3\right)\left(x+4\right)\left(x+6\right)} and \frac{5\left(x+6\right)}{\left(x+3\right)\left(x+4\right)\left(x+6\right)} have the same denominator, add them by adding their numerators.

$$\frac{6\left(x+4\right)+5\left(x+6\right)}{\left(x+3\right)\left(x+4\right)\left(x+6\right)}$$

Do the multiplications in 6\left(x+4\right)+5\left(x+6\right).

$$\frac{6x+24+5x+30}{\left(x+3\right)\left(x+4\right)\left(x+6\right)}$$

Combine like terms in 6x+24+5x+30.

$$\frac{11x+54}{\left(x+3\right)\left(x+4\right)\left(x+6\right)}$$

Expand \left(x+3\right)\left(x+4\right)\left(x+6\right).

$$\frac{11x+54}{x^{3}+13x^{2}+54x+72}$$

Differentiate w.r.t. x

$$-\frac{22x^{3}+305x^{2}+1404x+2124}{\left(\left(x+3\right)\left(x+4\right)\left(x+6\right)\right)^{2}}$$