Solve 3/a^2+6a+9+7/a^2-9

$$\frac { 3 } { a ^ { 2 } + 6 a + 9 } + \frac { 7 } { a ^ { 2 } - 9 }$$

Evaluate

$$\frac{2\left(5a+6\right)}{\left(a-3\right)\left(a+3\right)^{2}}$$

Short Solution Steps

Factor a^{2}+6a+9. Factor a^{2}-9.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a+3\right)^{2} and \left(a-3\right)\left(a+3\right) is \left(a-3\right)\left(a+3\right)^{2}. Multiply \frac{3}{\left(a+3\right)^{2}} times \frac{a-3}{a-3}. Multiply \frac{7}{\left(a-3\right)\left(a+3\right)} times \frac{a+3}{a+3}.

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

Since \frac{3\left(a-3\right)}{\left(a-3\right)\left(a+3\right)^{2}} and \frac{7\left(a+3\right)}{\left(a-3\right)\left(a+3\right)^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 3\left(a-3\right)+7\left(a+3\right).

$$\frac{3a-9+7a+21}{\left(a-3\right)\left(a+3\right)^{2}}$$

Combine like terms in 3a-9+7a+21.

$$\frac{10a+12}{\left(a-3\right)\left(a+3\right)^{2}}$$

Expand \left(a-3\right)\left(a+3\right)^{2}.

$$\frac{10a+12}{a^{3}+3a^{2}-9a-27}$$

Differentiate w.r.t. a

$$-\frac{2\left(10a^{2}+3a+27\right)}{\left(a-3\right)^{2}\left(a+3\right)^{3}}$$