Solve (1/x+3+6/x^2-9)+1/x^2-6x+914

$$( \frac { 1 } { x + 3 } + \frac { 6 } { x ^ { 2 } - 9 } ) + \frac { 1 } { x ^ { 2 } - 6 x + 9 } 14$$

Evaluate

$$\frac{x+11}{\left(x-3\right)^{2}}$$

Short Solution Steps

Express \frac{1}{x^{2}-6x+9}\times 14 as a single fraction.

$$\frac{1}{x+3}+\frac{6}{x^{2}-9}+\frac{14}{x^{2}-6x+9}$$

Factor x^{2}-9.

$$\frac{1}{x+3}+\frac{6}{\left(x-3\right)\left(x+3\right)}+\frac{14}{x^{2}-6x+9}$$

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

$$\frac{x-3}{\left(x-3\right)\left(x+3\right)}+\frac{6}{\left(x-3\right)\left(x+3\right)}+\frac{14}{x^{2}-6x+9}$$

Since \frac{x-3}{\left(x-3\right)\left(x+3\right)} and \frac{6}{\left(x-3\right)\left(x+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{x-3+6}{\left(x-3\right)\left(x+3\right)}+\frac{14}{x^{2}-6x+9}$$

Combine like terms in x-3+6.

$$\frac{x+3}{\left(x-3\right)\left(x+3\right)}+\frac{14}{x^{2}-6x+9}$$

Cancel out x+3 in both numerator and denominator.

$$\frac{1}{x-3}+\frac{14}{x^{2}-6x+9}$$

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

$$\frac{1}{x-3}+\frac{14}{\left(x-3\right)^{2}}$$

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

$$\frac{x-3}{\left(x-3\right)^{2}}+\frac{14}{\left(x-3\right)^{2}}$$

Since \frac{x-3}{\left(x-3\right)^{2}} and \frac{14}{\left(x-3\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{x-3+14}{\left(x-3\right)^{2}}$$

Combine like terms in x-3+14.

$$\frac{x+11}{\left(x-3\right)^{2}}$$

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

$$\frac{x+11}{x^{2}-6x+9}$$

Differentiate w.r.t. x

$$\frac{-x-25}{\left(x-3\right)^{3}}$$