1. **State the problem:** Solve the quadratic equation $$x^2 - |a-1| x - 1 = 0$$ for $x$. 2. **Recall the quadratic formula:** For an equation $$ax^2 + bx + c = 0$$, the solutions are given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a$, $b$, and $c$ are coefficients. 3. **Identify coefficients:** Here, $a=1$, $b = -|a-1|$, and $c = -1$. 4. **Apply the quadratic formula:** $$x = \frac{-(-|a-1|) \pm \sqrt{(-|a-1|)^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} = \frac{|a-1| \pm \sqrt{|a-1|^2 + 4}}{2}$$ 5. **Simplify inside the square root:** Since $|a-1|^2 = (a-1)^2$, $$x = \frac{|a-1| \pm \sqrt{(a-1)^2 + 4}}{2}$$ 6. **Final answer:** The two solutions for $x$ are $$x = \frac{|a-1| + \sqrt{(a-1)^2 + 4}}{2} \quad \text{and} \quad x = \frac{|a-1| - \sqrt{(a-1)^2 + 4}}{2}$$ These are the roots of the quadratic equation depending on the parameter $a$.