Question

Find the roots of the equation \( \frac{1}{x} - \frac{1}{x-2} = 3 \), \( x \neq 0,2 \).

Hint:

Roots of a Rational Equation: Convert fractions to quadratic equations and use the quadratic formula.

Answer 1

Rewriting the equation:
\[ \frac{(x-2) - x}{x(x-2)} = 3 \] \[ \frac{-2}{x(x-2)} = 3 \] \[ -2 = 3x(x-2) \] \[ 3x^2 - 6x + 2 = 0 \] Using the quadratic formula:
\[ x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(3)(2)}}{2(3)} \] \[ x = \frac{6 \pm \sqrt{36 - 24}}{6} \] \[ x = \frac{6 \pm \sqrt{12}}{6} \] \[ x = \frac{6 \pm 2\sqrt{3}}{6} \] \[ x = \frac{3 \pm \sqrt{3}}{3} \] Thus, the roots are \( \mathbf{\frac{3+\sqrt{3}}{3}, \frac{3-\sqrt{3}}{3}} \).
Correct Answer: \( \frac{3+\sqrt{3}}{3}, \frac{3-\sqrt{3}}{3} \)