Solve x+1/x^2-6x+9-x-1/x^2-5

$$\frac { x + 1 } { x ^ { 2 } - 6 x + 9 } - \frac { x - 1 } { x ^ { 2 } - 5 }$$

Evaluate

$$\frac{4\left(2x^{2}-5x+1\right)}{\left(x-3\right)^{2}\left(x^{2}-5\right)}$$

Short Solution Steps

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

$$\frac{x+1}{\left(x-3\right)^{2}}-\frac{x-1}{x^{2}-5}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-3\right)^{2} and x^{2}-5 is \left(x-3\right)^{2}\left(x^{2}-5\right). Multiply \frac{x+1}{\left(x-3\right)^{2}} times \frac{x^{2}-5}{x^{2}-5}. Multiply \frac{x-1}{x^{2}-5} times \frac{\left(x-3\right)^{2}}{\left(x-3\right)^{2}}.

$$\frac{\left(x+1\right)\left(x^{2}-5\right)}{\left(x-3\right)^{2}\left(x^{2}-5\right)}-\frac{\left(x-1\right)\left(x-3\right)^{2}}{\left(x-3\right)^{2}\left(x^{2}-5\right)}$$

Since \frac{\left(x+1\right)\left(x^{2}-5\right)}{\left(x-3\right)^{2}\left(x^{2}-5\right)} and \frac{\left(x-1\right)\left(x-3\right)^{2}}{\left(x-3\right)^{2}\left(x^{2}-5\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x+1\right)\left(x^{2}-5\right)-\left(x-1\right)\left(x-3\right)^{2}}{\left(x-3\right)^{2}\left(x^{2}-5\right)}$$

Do the multiplications in \left(x+1\right)\left(x^{2}-5\right)-\left(x-1\right)\left(x-3\right)^{2}.

$$\frac{x^{3}-5x+x^{2}-5-x^{3}+6x^{2}-9x+x^{2}-6x+9}{\left(x-3\right)^{2}\left(x^{2}-5\right)}$$

Combine like terms in x^{3}-5x+x^{2}-5-x^{3}+6x^{2}-9x+x^{2}-6x+9.

$$\frac{-20x+8x^{2}+4}{\left(x-3\right)^{2}\left(x^{2}-5\right)}$$

Expand \left(x-3\right)^{2}\left(x^{2}-5\right).

$$\frac{-20x+8x^{2}+4}{x^{4}-6x^{3}+4x^{2}+30x-45}$$

Expand

$$\frac{4\left(2x^{2}-5x+1\right)}{\left(x^{2}-5\right)\left(x^{2}-6x+9\right)}$$

Short Solution Steps

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

$$\frac{x+1}{\left(x-3\right)^{2}}-\frac{x-1}{x^{2}-5}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-3\right)^{2} and x^{2}-5 is \left(x-3\right)^{2}\left(x^{2}-5\right). Multiply \frac{x+1}{\left(x-3\right)^{2}} times \frac{x^{2}-5}{x^{2}-5}. Multiply \frac{x-1}{x^{2}-5} times \frac{\left(x-3\right)^{2}}{\left(x-3\right)^{2}}.

$$\frac{\left(x+1\right)\left(x^{2}-5\right)}{\left(x-3\right)^{2}\left(x^{2}-5\right)}-\frac{\left(x-1\right)\left(x-3\right)^{2}}{\left(x-3\right)^{2}\left(x^{2}-5\right)}$$

Since \frac{\left(x+1\right)\left(x^{2}-5\right)}{\left(x-3\right)^{2}\left(x^{2}-5\right)} and \frac{\left(x-1\right)\left(x-3\right)^{2}}{\left(x-3\right)^{2}\left(x^{2}-5\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x+1\right)\left(x^{2}-5\right)-\left(x-1\right)\left(x-3\right)^{2}}{\left(x-3\right)^{2}\left(x^{2}-5\right)}$$

Do the multiplications in \left(x+1\right)\left(x^{2}-5\right)-\left(x-1\right)\left(x-3\right)^{2}.

$$\frac{x^{3}-5x+x^{2}-5-x^{3}+6x^{2}-9x+x^{2}-6x+9}{\left(x-3\right)^{2}\left(x^{2}-5\right)}$$

Combine like terms in x^{3}-5x+x^{2}-5-x^{3}+6x^{2}-9x+x^{2}-6x+9.

$$\frac{-20x+8x^{2}+4}{\left(x-3\right)^{2}\left(x^{2}-5\right)}$$

Expand \left(x-3\right)^{2}\left(x^{2}-5\right).

$$\frac{-20x+8x^{2}+4}{x^{4}-6x^{3}+4x^{2}+30x-45}$$