Solve the equation: (60)/(x-5) - (6)/(x) = (1)/(6)

The given equation is:

$$\frac{60}{x-5} - \frac{6}{x} = \frac{1}{6}$$

Find a common denominator for the left side, which is $x(x-5)$:

$$\frac{60x - 6(x-5)}{x(x-5)} = \frac{1}{6}$$

Distribute the $-6$:

$$\frac{60x - 6x + 30}{x^2 - 5x} = \frac{1}{6}$$

Combine like terms in the numerator:

$$\frac{54x + 30}{x^2 - 5x} = \frac{1}{6}$$

Multiply both sides by $6(x^2 - 5x)$:

$$6(54x + 30) = 1(x^2 - 5x)$$

Distribute the 6 on the left side:

$$324x + 180 = x^2 - 5x$$

Move all terms to the right side to set the equation to zero:

$$0 = x^2 - 5x - 324x - 180$$

Combine like terms:

$$x^2 - 329x - 180 = 0$$

Move the constant term to the right side:

$$x^2 - 329x = 180$$

To complete the square, add $(\frac{b}{2})^2$ to both sides. Here, $b = -329$.
So, $(\frac{-329}{2})^2 = \frac{108241}{4}$.

$$x^2 - 329x + \left(\frac{-329}{2}\right)^2 = 180 + \left(\frac{-329}{2}\right)^2$$ $$\left(x - \frac{329}{2}\right)^2 = 180 + \frac{108241}{4}$$

Convert 180 to a fraction with denominator 4: $180 = \frac{180 \times 4}{4} = \frac{720}{4}$.

$$\left(x - \frac{329}{2}\right)^2 = \frac{720}{4} + \frac{108241}{4}$$ $$\left(x - \frac{329}{2}\right)^2 = \frac{720 + 108241}{4}$$ $$\left(x - \frac{329}{2}\right)^2 = \frac{108961}{4}$$

Take the square root of both sides:

$$x - \frac{329}{2} = \pm\sqrt{\frac{108961}{4}}$$ $$x - \frac{329}{2} = \pm\frac{\sqrt{108961}}{\sqrt{4}}$$ $$x - \frac{329}{2} = \pm\frac{\sqrt{108961}}{2}$$

Solve for $x$:

$$x = \frac{329}{2} \pm \frac{\sqrt{108961}}{2}$$ $$x = \frac{329 \pm \sqrt{108961}}{2}$$

The two solutions are:

$$x_1 = \frac{329 + \sqrt{108961}}{2}$$ $$x_2 = \frac{329 - \sqrt{108961}}{2}$$