1. **Problem Statement:** Solve the quadratic equation $$x^2 - 5x + 6 = 0$$. 2. **Formula Used:** The quadratic formula is $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where the quadratic equation is in the form $$ax^2 + bx + c = 0$$. 3. **Identify coefficients:** Here, $$a = 1$$, $$b = -5$$, and $$c = 6$$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-5)^2 - 4 \times 1 \times 6 = 25 - 24 = 1$$. 5. **Evaluate the roots:** $$x = \frac{-(-5) \pm \sqrt{1}}{2 \times 1} = \frac{5 \pm 1}{2}$$ 6. **Find the two solutions:** - $$x_1 = \frac{5 + 1}{2} = 3$$ - $$x_2 = \frac{5 - 1}{2} = 2$$ 7. **Interpretation:** The solutions $$x=2$$ and $$x=3$$ are the points where the quadratic expression equals zero. This method is fundamental in algebra and widely used in various fields including engineering and physics. This approach respects the university level rigor and can be appreciated in the context of Omani culture by emphasizing the importance of education and problem-solving skills in advancing knowledge and technology.