1. **State the problem:** Solve the quadratic equation $x^2 - x = 0$. 2. **Formula and rules:** To solve quadratic equations, we can factorize the expression and use the zero product property which states if $ab=0$, then either $a=0$ or $b=0$. 3. **Factorize the equation:** $$x^2 - x = x(x - 1) = 0$$ 4. **Apply zero product property:** Set each factor equal to zero: $$x = 0$$ $$x - 1 = 0$$ 5. **Solve each equation:** $$x = 0$$ $$x = 1$$ 6. **Final answer:** The solutions to the equation $x^2 - x = 0$ are $x = 0$ and $x = 1$.