Solve x/x^2-9-1/x^2-6x+9

$$\frac { x } { x ^ { 2 } - 9 } - \frac { 1 } { x ^ { 2 } - 6 x + 9 }$$

Evaluate

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

Short Solution Steps

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

$$\frac{x}{\left(x-3\right)\left(x+3\right)}-\frac{1}{\left(x-3\right)^{2}}$$

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{x}{\left(x-3\right)\left(x+3\right)} times \frac{x-3}{x-3}. Multiply \frac{1}{\left(x-3\right)^{2}} times \frac{x+3}{x+3}.

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

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

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

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

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

Combine like terms in x^{2}-3x-x-3.

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

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

$$\frac{x^{2}-4x-3}{x^{3}-3x^{2}-9x+27}$$

Differentiate w.r.t. x

$$\frac{45+3x+5x^{2}-x^{3}}{\left(x+3\right)^{2}\left(x-3\right)^{3}}$$