Solve 1/x^2-5x+6+1/x^2-3x+2+2/x^2-8x+15

$$\frac { 1 } { x ^ { 2 } - 5 x + 6 } + \frac { 1 } { x ^ { 2 } - 3 x + 2 } + \frac { 2 } { x ^ { 2 } - 8 x + 15 }$$

Evaluate

$$\frac{4}{\left(x-5\right)\left(x-1\right)}$$

Short Solution Steps

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

$$\frac{1}{\left(x-3\right)\left(x-2\right)}+\frac{1}{\left(x-2\right)\left(x-1\right)}+\frac{2}{x^{2}-8x+15}$$

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

$$\frac{x-1}{\left(x-3\right)\left(x-2\right)\left(x-1\right)}+\frac{x-3}{\left(x-3\right)\left(x-2\right)\left(x-1\right)}+\frac{2}{x^{2}-8x+15}$$

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

$$\frac{x-1+x-3}{\left(x-3\right)\left(x-2\right)\left(x-1\right)}+\frac{2}{x^{2}-8x+15}$$

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

$$\frac{2x-4}{\left(x-3\right)\left(x-2\right)\left(x-1\right)}+\frac{2}{x^{2}-8x+15}$$

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

$$\frac{2\left(x-2\right)}{\left(x-3\right)\left(x-2\right)\left(x-1\right)}+\frac{2}{x^{2}-8x+15}$$

Cancel out x-2 in both numerator and denominator.

$$\frac{2}{\left(x-3\right)\left(x-1\right)}+\frac{2}{x^{2}-8x+15}$$

Factor x^{2}-8x+15.

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

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

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

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

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

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

$$\frac{2x-10+2x-2}{\left(x-5\right)\left(x-3\right)\left(x-1\right)}$$

Combine like terms in 2x-10+2x-2.

$$\frac{4x-12}{\left(x-5\right)\left(x-3\right)\left(x-1\right)}$$

Factor the expressions that are not already factored in \frac{4x-12}{\left(x-5\right)\left(x-3\right)\left(x-1\right)}.

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

Cancel out x-3 in both numerator and denominator.

$$\frac{4}{\left(x-5\right)\left(x-1\right)}$$

Expand \left(x-5\right)\left(x-1\right).

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

Differentiate w.r.t. x

$$\frac{8\left(3-x\right)}{\left(\left(x-5\right)\left(x-1\right)\right)^{2}}$$