The function p is defined by \(\mathrm{p(n) = 50 - 2n^2}\). For which of the following values of n is...

1. TRANSLATE the problem information

• Given: \(\mathrm{p(n) = 50 - 2n^2}\) and we need \(\mathrm{p(n) = 18}\)
• This means: \(\mathrm{50 - 2n^2 = 18}\)

2. SIMPLIFY to solve for n

• Start with: \(\mathrm{50 - 2n^2 = 18}\)
• Subtract 50 from both sides: \(\mathrm{-2n^2 = 18 - 50 = -32}\)
• Divide both sides by -2: \(\mathrm{n^2 = 16}\)
• Take the square root: \(\mathrm{n = ±4}\)

3. Select the answer from the given choices

• We found \(\mathrm{n = ±4}\)
• Looking at choices (A) 2, (B) 4, (C) 6, (D) 8
• Only \(\mathrm{n = 4}\) appears, so the answer is (B)

4. Verify the answer

• Check: \(\mathrm{p(4) = 50 - 2(4^2) = 50 - 2(16) = 50 - 32 = 18}\) ✓

Answer: (B) 4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students often make sign errors or arithmetic mistakes during the multi-step algebraic process.

For example, when solving \(\mathrm{-2n^2 = -32}\), they might:

• Forget the negative signs and get \(\mathrm{n^2 = -16}\) (impossible)
• Make division errors and get \(\mathrm{n^2 = 64}\), leading to \(\mathrm{n = 8}\)
• Make arithmetic errors in the subtraction: \(\mathrm{18 - 50 = 32}\) instead of \(\mathrm{-32}\)

This may lead them to select Choice (D) (8) or get confused and guess.

The Bottom Line:

This problem tests careful algebraic manipulation more than complex reasoning. Students who rush through the arithmetic steps or don't track negative signs properly will select incorrect answers, even though they understand the basic setup.