1. **State the problem:** Solve the quadratic equation $$x^2 - 5x + 6 = 0$$. 2. **Formula and rules:** To solve a quadratic equation of the form $$ax^2 + bx + c = 0$$, we can factorize it if possible or use the quadratic formula $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$. 3. **Factorization:** Here, $$a=1$$, $$b=-5$$, and $$c=6$$. We look for two numbers that multiply to $$6$$ and add to $$-5$$. These numbers are $$-2$$ and $$-3$$. 4. **Rewrite the equation:** $$x^2 - 5x + 6 = (x - 2)(x - 3) = 0$$. 5. **Solve for x:** Set each factor equal to zero: - $$x - 2 = 0 \Rightarrow x = 2$$ - $$x - 3 = 0 \Rightarrow x = 3$$ 6. **Answer:** The solutions to the equation are $$x = 2$$ and $$x = 3$$.