1. **State the problem:** Solve the quadratic equation $$x^2 - 4x - 3 = 0$$ for $x$. 2. **Recall the quadratic formula:** For any quadratic equation $$ax^2 + bx + c = 0$$, the solutions for $x$ are given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a$, $b$, and $c$ are coefficients. 3. **Identify coefficients:** Here, $a = 1$, $b = -4$, and $c = -3$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-4)^2 - 4 \times 1 \times (-3) = 16 + 12 = 28$$ 5. **Apply the quadratic formula:** $$x = \frac{-(-4) \pm \sqrt{28}}{2 \times 1} = \frac{4 \pm \sqrt{28}}{2}$$ 6. **Simplify the square root:** $$\sqrt{28} = \sqrt{4 \times 7} = 2\sqrt{7}$$ 7. **Substitute back:** $$x = \frac{4 \pm 2\sqrt{7}}{2}$$ 8. **Simplify the fraction by canceling 2:** $$x = \frac{\cancel{2} \times 2 \pm \cancel{2} \times \sqrt{7}}{\cancel{2} \times 1} = 2 \pm \sqrt{7}$$ 9. **Final solutions:** $$x = 2 + \sqrt{7} \quad \text{or} \quad x = 2 - \sqrt{7}$$ These are the two roots of the quadratic equation.