Solve x/x^2-3x+2+1/x^2-5x+6

$$\frac { x } { x ^ { 2 } - 3 x + 2 } + \frac { 1 } { x ^ { 2 } - 5 x + 6 }$$

Evaluate

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

Short Solution Steps

Factor x^{2}-3x+2. Factor x^{2}-5x+6.

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

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

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

Since \frac{x\left(x-3\right)}{\left(x-3\right)\left(x-2\right)\left(x-1\right)} and \frac{x-1}{\left(x-3\right)\left(x-2\right)\left(x-1\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in x\left(x-3\right)+x-1.

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

Combine like terms in x^{2}-3x+x-1.

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

Expand \left(x-3\right)\left(x-2\right)\left(x-1\right).

$$\frac{x^{2}-2x-1}{x^{3}-6x^{2}+11x-6}$$

Differentiate w.r.t. x

$$\frac{23-24x+2x^{2}+4x^{3}-x^{4}}{\left(\left(x-3\right)\left(x-2\right)\left(x-1\right)\right)^{2}}$$