Solve 1/x^2-3x+2+1/x^2-5x+6+1/x^2-4x+3

$$\frac { 1 } { x ^ { 2 } - 3 x + 2 } + \frac { 1 } { x ^ { 2 } - 5 x + 6 } + \frac { 1 } { x ^ { 2 } - 4 x + 3 }$$

Evaluate

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

Short Solution Steps

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

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

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{1}{\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}.

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

Since \frac{x-3}{\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.

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

Combine like terms in x-3+x-1.

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

Factor the expressions that are not already factored in \frac{2x-4}{\left(x-3\right)\left(x-2\right)\left(x-1\right)}.

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

Cancel out x-2 in both numerator and denominator.

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

Factor x^{2}-4x+3.

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

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

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

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

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

Differentiate w.r.t. x

$$\frac{6\left(2-x\right)}{\left(\left(x-3\right)\left(x-1\right)\right)^{2}}$$