Solve x+6/x+1/x+6/frac{3{x}+x/x+6}

$$\frac { \frac { x + 6 } { x } + \frac { 1 } { x + 6 } } { \frac { 3 } { x } + \frac { x } { x + 6 } }$$

Evaluate

$$\frac{\left(x+4\right)\left(x+9\right)}{x^{2}+3x+18}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and x+6 is x\left(x+6\right). Multiply \frac{x+6}{x} times \frac{x+6}{x+6}. Multiply \frac{1}{x+6} times \frac{x}{x}.

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

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

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

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

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

Combine like terms in x^{2}+6x+6x+36+x.

$$\frac{\frac{x^{2}+13x+36}{x\left(x+6\right)}}{\frac{3}{x}+\frac{x}{x+6}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and x+6 is x\left(x+6\right). Multiply \frac{3}{x} times \frac{x+6}{x+6}. Multiply \frac{x}{x+6} times \frac{x}{x}.

$$\frac{\frac{x^{2}+13x+36}{x\left(x+6\right)}}{\frac{3\left(x+6\right)}{x\left(x+6\right)}+\frac{xx}{x\left(x+6\right)}}$$

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

$$\frac{\frac{x^{2}+13x+36}{x\left(x+6\right)}}{\frac{3\left(x+6\right)+xx}{x\left(x+6\right)}}$$

Do the multiplications in 3\left(x+6\right)+xx.

$$\frac{\frac{x^{2}+13x+36}{x\left(x+6\right)}}{\frac{3x+18+x^{2}}{x\left(x+6\right)}}$$

Divide \frac{x^{2}+13x+36}{x\left(x+6\right)} by \frac{3x+18+x^{2}}{x\left(x+6\right)} by multiplying \frac{x^{2}+13x+36}{x\left(x+6\right)} by the reciprocal of \frac{3x+18+x^{2}}{x\left(x+6\right)}.

$$\frac{\left(x^{2}+13x+36\right)x\left(x+6\right)}{x\left(x+6\right)\left(3x+18+x^{2}\right)}$$

Cancel out x\left(x+6\right) in both numerator and denominator.

$$\frac{x^{2}+13x+36}{x^{2}+3x+18}$$

Expand

$$\frac{x^{2}+13x+36}{x^{2}+3x+18}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and x+6 is x\left(x+6\right). Multiply \frac{x+6}{x} times \frac{x+6}{x+6}. Multiply \frac{1}{x+6} times \frac{x}{x}.

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

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

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

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

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

Combine like terms in x^{2}+6x+6x+36+x.

$$\frac{\frac{x^{2}+13x+36}{x\left(x+6\right)}}{\frac{3}{x}+\frac{x}{x+6}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and x+6 is x\left(x+6\right). Multiply \frac{3}{x} times \frac{x+6}{x+6}. Multiply \frac{x}{x+6} times \frac{x}{x}.

$$\frac{\frac{x^{2}+13x+36}{x\left(x+6\right)}}{\frac{3\left(x+6\right)}{x\left(x+6\right)}+\frac{xx}{x\left(x+6\right)}}$$

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

$$\frac{\frac{x^{2}+13x+36}{x\left(x+6\right)}}{\frac{3\left(x+6\right)+xx}{x\left(x+6\right)}}$$

Do the multiplications in 3\left(x+6\right)+xx.

$$\frac{\frac{x^{2}+13x+36}{x\left(x+6\right)}}{\frac{3x+18+x^{2}}{x\left(x+6\right)}}$$

Divide \frac{x^{2}+13x+36}{x\left(x+6\right)} by \frac{3x+18+x^{2}}{x\left(x+6\right)} by multiplying \frac{x^{2}+13x+36}{x\left(x+6\right)} by the reciprocal of \frac{3x+18+x^{2}}{x\left(x+6\right)}.

$$\frac{\left(x^{2}+13x+36\right)x\left(x+6\right)}{x\left(x+6\right)\left(3x+18+x^{2}\right)}$$

Cancel out x\left(x+6\right) in both numerator and denominator.

$$\frac{x^{2}+13x+36}{x^{2}+3x+18}$$