1. **State the problem:** Solve the equation $$\frac{x^2}{5} - x = \frac{1}{5}$$ for $x$. 2. **Rewrite the equation:** Multiply both sides by 5 to clear the denominators: $$x^2 - 5x = 1$$ 3. **Bring all terms to one side:** $$x^2 - 5x - 1 = 0$$ 4. **Use the quadratic formula:** For an equation $ax^2 + bx + c = 0$, the solutions are $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Here, $a=1$, $b=-5$, and $c=-1$. 5. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-5)^2 - 4(1)(-1) = 25 + 4 = 29$$ 6. **Find the roots:** $$x = \frac{-(-5) \pm \sqrt{29}}{2(1)} = \frac{5 \pm \sqrt{29}}{2}$$ 7. **Final answer:** $$x = \frac{5 + \sqrt{29}}{2}, \frac{5 - \sqrt{29}}{2}$$ These are the exact solutions in simplest form.