1. **State the problem:** Solve the equation $$\frac{1}{20} = \frac{1}{90 - x} + \frac{1}{x}$$ for $x$. 2. **Use the formula:** To solve equations involving sums of fractions, find a common denominator and combine terms. 3. **Find common denominator:** The denominators are $20$, $90 - x$, and $x$. Multiply both sides by the common denominator $20x(90 - x)$ to clear fractions: $$20x(90 - x) \times \frac{1}{20} = 20x(90 - x) \times \frac{1}{90 - x} + 20x(90 - x) \times \frac{1}{x}$$ 4. **Simplify each term:** $$x(90 - x) = 20x + 20(90 - x)$$ 5. **Expand terms:** $$90x - x^2 = 20x + 1800 - 20x$$ 6. **Simplify right side:** $$90x - x^2 = 1800$$ 7. **Rearrange to standard quadratic form:** $$-x^2 + 90x - 1800 = 0$$ Multiply both sides by $-1$ to simplify: $$\cancel{-}x^2 + \cancel{90}x - \cancel{1800} = \cancel{0}$$ becomes $$x^2 - 90x + 1800 = 0$$ 8. **Solve quadratic equation:** Use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=-90$, $c=1800$. 9. **Calculate discriminant:** $$\Delta = (-90)^2 - 4 \times 1 \times 1800 = 8100 - 7200 = 900$$ 10. **Find roots:** $$x = \frac{90 \pm \sqrt{900}}{2} = \frac{90 \pm 30}{2}$$ 11. **Calculate each root:** - $$x_1 = \frac{90 + 30}{2} = \frac{120}{2} = 60$$ - $$x_2 = \frac{90 - 30}{2} = \frac{60}{2} = 30$$ 12. **Check for restrictions:** $x$ cannot be $0$ or $90$ because of denominators. 13. **Final answer:** $$x = 30 \text{ or } x = 60$$