1. **State the problem:** Solve the quadratic equation $$x^2 - 11x + 28 = 0$$. 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, $$a = 1$$, $$b = -11$$, and $$c = 28$$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-11)^2 - 4 \times 1 \times 28 = 121 - 112 = 9$$. 5. **Find the square root of the discriminant:** $$\sqrt{\Delta} = \sqrt{9} = 3$$. 6. **Apply the quadratic formula:** $$x = \frac{-(-11) \pm 3}{2 \times 1} = \frac{11 \pm 3}{2}$$. 7. **Calculate the two solutions:** - For the plus sign: $$x = \frac{11 + 3}{2} = \frac{14}{2} = 7$$. - For the minus sign: $$x = \frac{11 - 3}{2} = \frac{8}{2} = 4$$. 8. **Final answer:** The solutions to the equation are $$x = 7$$ and $$x = 4$$.