Solve (x+9/x-6):3x^2-18x+27/x^2-36

$$( x + \frac { 9 } { x - 6 } ) : \frac { 3 x ^ { 2 } - 18 x + 27 } { x ^ { 2 } - 36 }$$

Evaluate

$$\frac{x}{3}+2$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{x-6}{x-6}.

$$\frac{\frac{x\left(x-6\right)}{x-6}+\frac{9}{x-6}}{\frac{3x^{2}-18x+27}{x^{2}-36}}$$

Since \frac{x\left(x-6\right)}{x-6} and \frac{9}{x-6} have the same denominator, add them by adding their numerators.

$$\frac{\frac{x\left(x-6\right)+9}{x-6}}{\frac{3x^{2}-18x+27}{x^{2}-36}}$$

Do the multiplications in x\left(x-6\right)+9.

$$\frac{\frac{x^{2}-6x+9}{x-6}}{\frac{3x^{2}-18x+27}{x^{2}-36}}$$

Divide \frac{x^{2}-6x+9}{x-6} by \frac{3x^{2}-18x+27}{x^{2}-36} by multiplying \frac{x^{2}-6x+9}{x-6} by the reciprocal of \frac{3x^{2}-18x+27}{x^{2}-36}.

$$\frac{\left(x^{2}-6x+9\right)\left(x^{2}-36\right)}{\left(x-6\right)\left(3x^{2}-18x+27\right)}$$

Factor the expressions that are not already factored.

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

Cancel out \left(x-6\right)\left(x-3\right)^{2} in both numerator and denominator.

$$\frac{x+6}{3}$$

Expand

$$\frac{x}{3}+2$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{x-6}{x-6}.

$$\frac{\frac{x\left(x-6\right)}{x-6}+\frac{9}{x-6}}{\frac{3x^{2}-18x+27}{x^{2}-36}}$$

Since \frac{x\left(x-6\right)}{x-6} and \frac{9}{x-6} have the same denominator, add them by adding their numerators.

$$\frac{\frac{x\left(x-6\right)+9}{x-6}}{\frac{3x^{2}-18x+27}{x^{2}-36}}$$

Do the multiplications in x\left(x-6\right)+9.

$$\frac{\frac{x^{2}-6x+9}{x-6}}{\frac{3x^{2}-18x+27}{x^{2}-36}}$$

Divide \frac{x^{2}-6x+9}{x-6} by \frac{3x^{2}-18x+27}{x^{2}-36} by multiplying \frac{x^{2}-6x+9}{x-6} by the reciprocal of \frac{3x^{2}-18x+27}{x^{2}-36}.

$$\frac{\left(x^{2}-6x+9\right)\left(x^{2}-36\right)}{\left(x-6\right)\left(3x^{2}-18x+27\right)}$$

Factor the expressions that are not already factored.

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

Cancel out \left(x-6\right)\left(x-3\right)^{2} in both numerator and denominator.

$$\frac{x+6}{3}$$