Solve 4/x^2-6x+9-5/x^2-9

$$\frac { 4 } { x ^ { 2 } - 6 x + 9 } - \frac { 5 } { x ^ { 2 } - 9 }$$

Evaluate

$$\frac{27-x}{\left(x+3\right)\left(x-3\right)^{2}}$$

Short Solution Steps

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

$$\frac{4}{\left(x-3\right)^{2}}-\frac{5}{\left(x-3\right)\left(x+3\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-3\right)^{2} and \left(x-3\right)\left(x+3\right) is \left(x+3\right)\left(x-3\right)^{2}. Multiply \frac{4}{\left(x-3\right)^{2}} times \frac{x+3}{x+3}. Multiply \frac{5}{\left(x-3\right)\left(x+3\right)} times \frac{x-3}{x-3}.

$$\frac{4\left(x+3\right)}{\left(x+3\right)\left(x-3\right)^{2}}-\frac{5\left(x-3\right)}{\left(x+3\right)\left(x-3\right)^{2}}$$

Since \frac{4\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.

$$\frac{4\left(x+3\right)-5\left(x-3\right)}{\left(x+3\right)\left(x-3\right)^{2}}$$

Do the multiplications in 4\left(x+3\right)-5\left(x-3\right).

$$\frac{4x+12-5x+15}{\left(x+3\right)\left(x-3\right)^{2}}$$

Combine like terms in 4x+12-5x+15.

$$\frac{-x+27}{\left(x+3\right)\left(x-3\right)^{2}}$$

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

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

Differentiate w.r.t. x

$$\frac{\left(2x-39\right)^{2}-1665}{2\left(x+3\right)^{2}\left(x-3\right)^{3}}$$