1. **State the problem:** Solve the quadratic equation $$f(x) = x^2 - 5x + 6 = 0$$ to find its roots. 2. **Recall the quadratic formula:** For any quadratic equation $$ax^2 + bx + c = 0$$, the roots are given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $$a=1$$, $$b=-5$$, and $$c=6$$ in this case. 3. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-5)^2 - 4(1)(6) = 25 - 24 = 1$$ Since $$\Delta > 0$$, there are two distinct real roots. 4. **Factorize the quadratic:** We look for two numbers that multiply to $$6$$ and add to $$-5$$, which are $$-2$$ and $$-3$$. So, $$x^2 - 5x + 6 = (x - 2)(x - 3)$$ 5. **Solve for roots:** Set each factor equal to zero: $$x - 2 = 0 \Rightarrow x = 2$$ $$x - 3 = 0 \Rightarrow x = 3$$ 6. **Verify with quadratic formula:** $$x = \frac{-(-5) \pm \sqrt{1}}{2(1)} = \frac{5 \pm 1}{2}$$ So, $$x = \frac{5 + 1}{2} = 3$$ $$x = \frac{5 - 1}{2} = 2$$ **Final answer:** The roots of the quadratic equation are $$x = 2$$ and $$x = 3$$.