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

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

Evaluate

$$\frac{3\left(x+13\right)}{\left(x-3\right)\left(x+3\right)^{2}}$$

Short Solution Steps

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

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

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

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

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

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

Do the multiplications in 8\left(x+3\right)-5\left(x-3\right).

$$\frac{8x+24-5x+15}{\left(x-3\right)\left(x+3\right)^{2}}$$

Combine like terms in 8x+24-5x+15.

$$\frac{3x+39}{\left(x-3\right)\left(x+3\right)^{2}}$$

Expand \left(x-3\right)\left(x+3\right)^{2}.

$$\frac{3x+39}{x^{3}+3x^{2}-9x-27}$$

Differentiate w.r.t. x

$$-\frac{6\left(\left(x+9\right)^{2}-96\right)}{\left(x-3\right)^{2}\left(x+3\right)^{3}}$$