1. **Problem:** Solve the equation $$\lfloor x \rfloor^2 - 5\lfloor x \rfloor + 6 = 0$$ where $$\lfloor x \rfloor$$ is the greatest integer function (floor function). Find the interval for $$x$$. 2. **Formula and rules:** The floor function $$\lfloor x \rfloor$$ returns the greatest integer less than or equal to $$x$$. Let $$n = \lfloor x \rfloor$$, then the equation becomes: $$n^2 - 5n + 6 = 0$$ 3. **Solve the quadratic equation:** $$n^2 - 5n + 6 = 0$$ Factorize: $$n^2 - 5n + 6 = (n - 2)(n - 3) = 0$$ So, $$n = 2 \quad \text{or} \quad n = 3$$ 4. **Interpretation:** Since $$n = \lfloor x \rfloor$$, $$x$$ lies in intervals where the floor is 2 or 3. - If $$\lfloor x \rfloor = 2$$, then $$2 \leq x < 3$$ - If $$\lfloor x \rfloor = 3$$, then $$3 \leq x < 4$$ 5. **Combine intervals:** $$x \in [2,3) \cup [3,4) = [2,4)$$ 6. **Answer:** The correct interval is $$x \in [2,4)$$ which corresponds to option (d).