1. **State the problem:** Solve the rational equation $$\frac{1}{x} + \frac{1}{x-2} = \frac{1}{4}$$ by simplifying it into a quadratic equation of the form $$x^2 - bx + c = 0$$. 2. **Find the least common denominator (LCD):** The denominators are $$x$$ and $$x-2$$, so the LCD is $$x(x-2)$$. 3. **Multiply both sides by the LCD to clear denominators:** $$x(x-2) \left( \frac{1}{x} + \frac{1}{x-2} \right) = x(x-2) \cdot \frac{1}{4}$$ 4. **Simplify each term:** $$ (x-2) + x = \frac{x(x-2)}{4} $$ 5. **Combine like terms on the left:** $$ 2x - 2 = \frac{x^2 - 2x}{4} $$ 6. **Multiply both sides by 4 to eliminate the fraction:** $$ 4(2x - 2) = x^2 - 2x $$ $$ 8x - 8 = x^2 - 2x $$ 7. **Bring all terms to one side to form a quadratic equation:** $$ 0 = x^2 - 2x - 8x + 8 $$ $$ 0 = x^2 - 10x + 8 $$ 8. **Identify coefficients:** The quadratic equation is $$x^2 - 10x + 8 = 0$$, so $$b = 10$$ and $$c = 8$$. **Final answer:** $$x^2 - 10x + 8 = 0$$