A positive real number n satisfies the equation n = 2sqrt(n) + 8. What is the value of n?

1. TRANSLATE the radical equation into a manageable form

• Given information: \(\mathrm{n = 2\sqrt{n} + 8}\), where \(\mathrm{n \gt 0}\)
• The presence of both \(\mathrm{n}\) and \(\mathrm{\sqrt{n}}\) makes this equation complex to solve directly
• INFER that substitution will simplify: Let \(\mathrm{k = \sqrt{n}}\), where \(\mathrm{k \gt 0}\)
• This means \(\mathrm{n = k^2}\), so our equation becomes: \(\mathrm{k^2 = 2k + 8}\)

2. SIMPLIFY by converting to standard quadratic form

• Rearrange: \(\mathrm{k^2 - 2k - 8 = 0}\)
• INFER that factoring is the most efficient approach here
• Look for two numbers that multiply to -8 and add to -2: those are -4 and +2
• Factor: \(\mathrm{(k - 4)(k + 2) = 0}\)

3. Solve the factored equation

• From \(\mathrm{(k - 4)(k + 2) = 0}\), we get \(\mathrm{k = 4}\) or \(\mathrm{k = -2}\)

4. APPLY CONSTRAINTS to select the valid solution

• Since \(\mathrm{k = \sqrt{n}}\) and \(\mathrm{n}\) is positive, \(\mathrm{k}\) must be positive
• Reject \(\mathrm{k = -2}\), accept \(\mathrm{k = 4}\)

5. Convert back to find n

• Since \(\mathrm{k = \sqrt{n} = 4}\), we have \(\mathrm{n = k^2 = 16}\)

Answer: D) 16

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students attempt to solve \(\mathrm{n = 2\sqrt{n} + 8}\) directly without substitution, leading to messy algebra with radicals that becomes very difficult to manipulate. They might try squaring both sides immediately, which creates a more complex equation: \(\mathrm{n^2 = (2\sqrt{n} + 8)^2 = 4n + 32\sqrt{n} + 64}\). This approach quickly becomes unmanageable and leads to confusion and guessing.

Second Most Common Error:

Poor APPLY CONSTRAINTS reasoning: Students correctly find \(\mathrm{k = 4}\) or \(\mathrm{k = -2}\) but fail to recognize that \(\mathrm{k = \sqrt{n}}\) must be positive. They might incorrectly accept \(\mathrm{k = -2}\), leading to \(\mathrm{n = (-2)^2 = 4}\). This would lead them to select Choice A (4), which doesn't satisfy the original equation when checked.

The Bottom Line:

This problem tests whether students recognize when substitution simplifies a complex equation and whether they properly apply domain restrictions. The key insight is that \(\mathrm{\sqrt{n}}\) notation always means the principal (positive) square root.