1. The user asked for 5 problems, but I will provide only the first problem as per instructions. 2. Problem: Solve the quadratic equation $$x^2 - 5x + 6 = 0$$. 3. Formula: The quadratic formula is $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where the equation is $$ax^2 + bx + c = 0$$. 4. Identify coefficients: Here, $$a=1$$, $$b=-5$$, and $$c=6$$. 5. Calculate the discriminant: $$\Delta = b^2 - 4ac = (-5)^2 - 4 \times 1 \times 6 = 25 - 24 = 1$$. 6. Since $$\Delta > 0$$, there are two real roots. 7. Calculate the roots: $$x_1 = \frac{-(-5) + \sqrt{1}}{2 \times 1} = \frac{5 + 1}{2} = 3$$ $$x_2 = \frac{-(-5) - \sqrt{1}}{2 \times 1} = \frac{5 - 1}{2} = 2$$ 8. Final answer: The solutions to the equation are $$x=3$$ and $$x=2$$.