https://file.aiquickdraw.com/mathbot/file/1726759993577mp3hf24z.jpg

The given equation is:

\[
\left\lfloor \sqrt{n} + \sqrt{n+1} + \sqrt{n+2} \right\rfloor = \left\lfloor \sqrt{9n + 8} \right\rfloor
\]

Where \( \lfloor x \rfloor \) represents the floor function, which gives the greatest integer less than or equal to \( x \).

To solve this equation, we need to explore the possible integer values of \( n \) such that the floor values of both sides are equal.

### Steps:

1. **Analyze the behavior of both sides**:
    - The left side involves three square roots added together.
    - The right side simplifies to a single square root function of \( n \).

2. **Approximate small values of \( n \)**:
    We will try integer values of \( n \) to see when the equation holds.

Let's start solving this through approximations:

#### For \( n = 0 \):
\[
\left\lfloor \sqrt{0} + \sqrt{1} + \sqrt{2} \right\rfloor = \left\lfloor 0 + 1 + 1.414 \right\rfloor = \left\lfloor 2.414 \right\rfloor = 2
\]
\[
\left\lfloor \sqrt{9(0) + 8} \right\rfloor = \left\lfloor \sqrt{8} \right\rfloor = \left\lfloor 2.828 \right\rfloor = 2
\]
Thus, \( n = 0 \) is a solution.

#### For \( n = 1 \):
\[
\left\lfloor \sqrt{1} + \sqrt{2} + \sqrt{3} \right\rfloor = \left\lfloor 1 + 1.414 + 1.732 \right\rfloor = \left\lfloor 4.146 \right\rfloor = 4
\]
\[
\left\lfloor \sqrt{9(1) + 8} \right\rfloor = \left\lfloor \sqrt{17} \right\rfloor = \left\lfloor 4.123 \right\rfloor = 4
\]
Thus, \( n = 1 \) is also a solution.

#### For \( n = 2 \):
\[
\left\lfloor \sqrt{2} + \sqrt{3} + \sqrt{4} \right\rfloor = \left\lfloor 1.414 + 1.732 + 2 \right\rfloor = \left\lfloor 5.146 \right\rfloor = 5
\]
\[
\left\lfloor \sqrt{9(2) + 8} \right\rfloor = \left\lfloor \sqrt{26} \right\rfloor = \left\lfloor 5.099 \right\rfloor = 5
\]
Thus, \( n = 2 \) is another solution.

### Conclusion:
The values of \( n = 0, 1, 2 \) satisfy the equation. Therefore, these are the solutions to the equation.

Let me know if you'd like further clarification, and here are 5 related questions to explore:

1. How does the floor function affect the solution process?
2. Why is it important to test small values of \( n \) first in equations like this?
3. Can this equation have solutions for larger values of \( n \)?
4. What methods can be used to approximate the values of square roots without a calculator?
5. How does the presence of square roots in the equation impact the behavior of the functions?

**Tip**: When dealing with floor functions and square roots, always check boundary values carefully to ensure accuracy in approximations.

Solve the equation: ⌊√n + √(n + 1) + √(n + 2)⌋ = ⌊√(9n + 8)⌋

Discover how to solve the equation involving the floor function and square roots: ⌊√n + √(n + 1) + √(n + 2)⌋ = ⌊√(9n + 8)⌋. This problem touches on key concepts like square roots, floor functions, and approximations. Suitable for students in Grades 10-12.

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