1. **State the problem:** Solve the equation $x^2 - 5x + 6 = 0$. 2. **Formula and rules:** To solve a quadratic equation $ax^2 + bx + c = 0$, we can factor it, complete the square, or use the quadratic formula. Here, factoring is straightforward. 3. **Factor the quadratic:** Find two numbers that multiply to $6$ and add to $-5$. These are $-2$ and $-3$. 4. **Write the factored form:** $$x^2 - 5x + 6 = (x - 2)(x - 3) = 0$$ 5. **Apply zero product property:** If $(x - 2)(x - 3) = 0$, then either $x - 2 = 0$ or $x - 3 = 0$. 6. **Solve each equation:** $$x - 2 = 0 \Rightarrow x = 2$$ $$x - 3 = 0 \Rightarrow x = 3$$ 7. **Final answer:** The solutions are $x = 2$ and $x = 3$.