1. **State the problem:** Solve the quadratic equation $$f(x) = x^2 - 2x - 3 = 0$$. 2. **Recall the quadratic formula:** For any quadratic equation $$ax^2 + bx + c = 0$$, the solutions are given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$. 3. **Identify coefficients:** Here, $$a = 1$$, $$b = -2$$, and $$c = -3$$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-2)^2 - 4 \times 1 \times (-3) = 4 + 12 = 16$$. 5. **Apply the quadratic formula:** $$x = \frac{-(-2) \pm \sqrt{16}}{2 \times 1} = \frac{2 \pm 4}{2}$$. 6. **Find the two solutions:** - For the plus sign: $$x = \frac{2 + 4}{2} = \frac{6}{2} = 3$$. - For the minus sign: $$x = \frac{2 - 4}{2} = \frac{\cancel{2} - 4}{\cancel{2} \times 1} = \frac{-2}{2} = -1$$. 7. **Final answer:** The solutions to the equation are $$x = 3$$ and $$x = -1$$.