Solve 7/a^2+10a+25+8/a^2-25

$$\frac { 7 } { a ^ { 2 } + 10 a + 25 } + \frac { 8 } { a ^ { 2 } - 25 }$$

Evaluate

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

Short Solution Steps

Factor a^{2}+10a+25. Factor a^{2}-25.

$$\frac{7}{\left(a+5\right)^{2}}+\frac{8}{\left(a-5\right)\left(a+5\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a+5\right)^{2} and \left(a-5\right)\left(a+5\right) is \left(a-5\right)\left(a+5\right)^{2}. Multiply \frac{7}{\left(a+5\right)^{2}} times \frac{a-5}{a-5}. Multiply \frac{8}{\left(a-5\right)\left(a+5\right)} times \frac{a+5}{a+5}.

$$\frac{7\left(a-5\right)}{\left(a-5\right)\left(a+5\right)^{2}}+\frac{8\left(a+5\right)}{\left(a-5\right)\left(a+5\right)^{2}}$$

Since \frac{7\left(a-5\right)}{\left(a-5\right)\left(a+5\right)^{2}} and \frac{8\left(a+5\right)}{\left(a-5\right)\left(a+5\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{7\left(a-5\right)+8\left(a+5\right)}{\left(a-5\right)\left(a+5\right)^{2}}$$

Do the multiplications in 7\left(a-5\right)+8\left(a+5\right).

$$\frac{7a-35+8a+40}{\left(a-5\right)\left(a+5\right)^{2}}$$

Combine like terms in 7a-35+8a+40.

$$\frac{15a+5}{\left(a-5\right)\left(a+5\right)^{2}}$$

Expand \left(a-5\right)\left(a+5\right)^{2}.

$$\frac{15a+5}{a^{3}+5a^{2}-25a-125}$$

Differentiate w.r.t. a

$$\frac{10\left(-3a^{2}+6a-35\right)}{\left(a-5\right)^{2}\left(a+5\right)^{3}}$$