Solve a/a^2+a-2+2/a^2-7a+6

$$\frac { a } { a ^ { 2 } + a - 2 } + \frac { 2 } { a ^ { 2 } - 7 a + 6 }$$

Evaluate

$$\frac{\left(a-2\right)^{2}}{\left(a-6\right)\left(a-1\right)\left(a+2\right)}$$

Short Solution Steps

Factor a^{2}+a-2. Factor a^{2}-7a+6.

$$\frac{a}{\left(a-1\right)\left(a+2\right)}+\frac{2}{\left(a-6\right)\left(a-1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a-1\right)\left(a+2\right) and \left(a-6\right)\left(a-1\right) is \left(a-6\right)\left(a-1\right)\left(a+2\right). Multiply \frac{a}{\left(a-1\right)\left(a+2\right)} times \frac{a-6}{a-6}. Multiply \frac{2}{\left(a-6\right)\left(a-1\right)} times \frac{a+2}{a+2}.

$$\frac{a\left(a-6\right)}{\left(a-6\right)\left(a-1\right)\left(a+2\right)}+\frac{2\left(a+2\right)}{\left(a-6\right)\left(a-1\right)\left(a+2\right)}$$

Since \frac{a\left(a-6\right)}{\left(a-6\right)\left(a-1\right)\left(a+2\right)} and \frac{2\left(a+2\right)}{\left(a-6\right)\left(a-1\right)\left(a+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{a\left(a-6\right)+2\left(a+2\right)}{\left(a-6\right)\left(a-1\right)\left(a+2\right)}$$

Do the multiplications in a\left(a-6\right)+2\left(a+2\right).

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

Combine like terms in a^{2}-6a+2a+4.

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

Expand \left(a-6\right)\left(a-1\right)\left(a+2\right).

$$\frac{a^{2}-4a+4}{a^{3}-5a^{2}-8a+12}$$

Differentiate w.r.t. a

$$\frac{\left(a-2\right)\left(8-28a+6a^{2}-a^{3}\right)}{\left(\left(a-6\right)\left(a-1\right)\left(a+2\right)\right)^{2}}$$