Solve 3/a^2-4-2/a^2-4a+4

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

Evaluate

$$\frac{a-10}{\left(a+2\right)\left(a-2\right)^{2}}$$

Short Solution Steps

Factor a^{2}-4. Factor a^{2}-4a+4.

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

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

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

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

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

Do the multiplications in 3\left(a-2\right)-2\left(a+2\right).

$$\frac{3a-6-2a-4}{\left(a+2\right)\left(a-2\right)^{2}}$$

Combine like terms in 3a-6-2a-4.

$$\frac{a-10}{\left(a+2\right)\left(a-2\right)^{2}}$$

Expand \left(a+2\right)\left(a-2\right)^{2}.

$$\frac{a-10}{a^{3}-2a^{2}-4a+8}$$

Differentiate w.r.t. a

$$-\frac{2\left(\left(a-7\right)^{2}-57\right)}{\left(a+2\right)^{2}\left(a-2\right)^{3}}$$