The expression (6x^2 + ax)/2x is equivalent to 3x + 5 for some constant a. What is the value of...

1. SIMPLIFY the complex fraction

• Given: \(\frac{6x^2 + ax}{2x}\) is equivalent to \(3x + 5\)
• Break the fraction apart by dividing each term in the numerator by \(2x\):
  • \(\frac{6x^2}{2x} = 3x\) (the \(\frac{x^2}{x} = x\), and \(\frac{6}{2} = 3\))
  • \(\frac{ax}{2x} = \frac{a}{2}\) (the x terms cancel, leaving \(\frac{a}{2}\))
• So \(\frac{6x^2 + ax}{2x} = 3x + \frac{a}{2}\)

2. INFER the equivalence condition

• We now have: \(3x + \frac{a}{2} = 3x + 5\)
• For two algebraic expressions to be equivalent, the coefficients of like terms must be equal
• The x-terms already match: \(3x = 3x\) ✓
• The constant terms must also match: \(\frac{a}{2} = 5\)

3. SIMPLIFY to solve for a

• From \(\frac{a}{2} = 5\)
• Multiply both sides by 2: \(a = 10\)

Answer: C (10)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skills: Students incorrectly distribute the division, often writing:

\(\frac{6x^2 + ax}{2x} = 6x^2 + \frac{ax}{2x} = 6x^2 + \frac{a}{2}\)

They forget that the entire numerator must be divided by \(2x\), not just the second term. This leads to the wrong equation \(6x^2 + \frac{a}{2} = 3x + 5\), which has no solution since the \(x^2\) term doesn't match. This causes confusion and leads to guessing.

Second Most Common Error:

Inadequate INFER reasoning: Students correctly simplify to \(3x + \frac{a}{2} = 3x + 5\) but then try to solve this as an equation for x instead of recognizing it as an identity. They might subtract \(3x\) from both sides to get \(\frac{a}{2} = 5\) (correct) but second-guess themselves because "the x disappeared," leading them to make computational errors or select Choice A (6) through faulty reasoning.

The Bottom Line:

This problem tests whether students can properly handle algebraic fractions and understand what "equivalent expressions" means mathematically - not just that they're equal for some values, but identical in form.