Question:The operation © is defined for all numbers p and q by the equation p © q = (q^2 -...

1. TRANSLATE the problem information

• Given information:
  • Operation © is defined as: \(\mathrm{p © q = \frac{q^2 - 3p}{p - q}}\)
  • We need to find: \(\mathrm{3 © (-1)}\)
  • This means \(\mathrm{p = 3}\) and \(\mathrm{q = -1}\)

2. TRANSLATE the specific calculation

• Substitute our values into the formula:
  • \(\mathrm{p © q = \frac{q^2 - 3p}{p - q}}\)
  • \(\mathrm{3 © (-1) = \frac{(-1)^2 - 3(3)}{3 - (-1)}}\)

3. SIMPLIFY the numerator

• Calculate \(\mathrm{(-1)^2 - 3(3)}\):
  • \(\mathrm{(-1)^2 = 1}\) (negative squared gives positive)
  • \(\mathrm{3(3) = 9}\)
  • So: \(\mathrm{1 - 9 = -8}\)

4. SIMPLIFY the denominator

• Calculate \(\mathrm{3 - (-1)}\):
  • Subtracting a negative is the same as adding: \(\mathrm{3 - (-1) = 3 + 1 = 4}\)

5. SIMPLIFY the final division

• Complete the calculation: \(\mathrm{\frac{-8}{4} = -2}\)

Answer: (C) -2

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution with negative numbers: Students often make sign errors when working with \(\mathrm{(-1)^2}\) or when calculating \(\mathrm{3 - (-1)}\).

A common mistake is thinking \(\mathrm{(-1)^2 = -1}\) instead of \(\mathrm{+1}\), which would make the numerator \(\mathrm{(-1) - 9 = -10}\) instead of \(\mathrm{-8}\). This leads to \(\mathrm{\frac{-10}{4} = -2.5}\), which isn't among the answer choices and causes confusion.

Another frequent error is calculating \(\mathrm{3 - (-1)}\) as \(\mathrm{2}\) instead of \(\mathrm{4}\), treating it like \(\mathrm{3 - 1}\). This gives \(\mathrm{\frac{-8}{2} = -4}\), leading them to select Choice (B) (-4).

Second Most Common Error:

Poor TRANSLATE reasoning: Students sometimes mix up which variable is p and which is q when substituting into the formula.

If they incorrectly use \(\mathrm{p = -1}\) and \(\mathrm{q = 3}\), they get:

\(\mathrm{(-1) © 3 = \frac{3^2 - 3(-1)}{-1 - 3}}\)

\(\mathrm{= \frac{9 + 3}{-4}}\)

\(\mathrm{= \frac{12}{-4}}\)

\(\mathrm{= -3}\)

Since -3 isn't an option, this leads to confusion and guessing.

The Bottom Line:

Custom operations require careful substitution and meticulous arithmetic with negative numbers. The key is to work slowly through each step, double-checking signs at every stage.