Solve (1/x+3+6/x^2-9)div1/x^2-6x+9

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

Evaluate

$$x-3$$

Short Solution Steps

Factor x^{2}-9.

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

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}.

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

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.

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

Combine like terms in x-3+6.

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

Cancel out x+3 in both numerator and denominator.

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

Divide \frac{1}{x-3} by \frac{1}{x^{2}-6x+9} by multiplying \frac{1}{x-3} by the reciprocal of \frac{1}{x^{2}-6x+9}.

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

Factor the expressions that are not already factored.

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

Cancel out x-3 in both numerator and denominator.

$$x-3$$

Expand

$$x-3$$

Short Solution Steps

Factor x^{2}-9.

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

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}.

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

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.

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

Combine like terms in x-3+6.

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

Cancel out x+3 in both numerator and denominator.

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

Divide \frac{1}{x-3} by \frac{1}{x^{2}-6x+9} by multiplying \frac{1}{x-3} by the reciprocal of \frac{1}{x^{2}-6x+9}.

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

Factor the expressions that are not already factored.

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

Cancel out x-3 in both numerator and denominator.

$$x-3$$