The expression 2x^2 + ax is equivalent to \(\mathrm{x}(2\mathrm{x} + 7)\) for some constant a. What is the value of...

1. TRANSLATE the problem information

• Given information:
  • \(2\mathrm{x}^2 + \mathrm{ax}\) is equivalent to \(\mathrm{x}(2\mathrm{x} + 7)\)
  • Need to find the value of constant a
• What this tells us: The two expressions must be equal for all values of x

2. SIMPLIFY the factored expression

• Expand \(\mathrm{x}(2\mathrm{x} + 7)\) using the distributive property:
  • \(\mathrm{x}(2\mathrm{x} + 7) = \mathrm{x} \cdot 2\mathrm{x} + \mathrm{x} \cdot 7\)
  • \(= 2\mathrm{x}^2 + 7\mathrm{x}\)

3. INFER the coefficient relationship

• Now we can compare the two polynomial expressions:
  • Original form: \(2\mathrm{x}^2 + \mathrm{ax}\)
  • Expanded form: \(2\mathrm{x}^2 + 7\mathrm{x}\)
• Since these expressions are equivalent, their coefficients must match:
  • Coefficient of \(\mathrm{x}^2\): \(2 = 2\) ✓
  • Coefficient of x: \(\mathrm{a} = 7\)

Answer: D. 7

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make mistakes when applying the distributive property

They might expand \(\mathrm{x}(2\mathrm{x} + 7)\) incorrectly as:

• \(\mathrm{x}(2\mathrm{x} + 7) = 2\mathrm{x}^2 + 7\) (forgetting to multiply 7 by x)
• OR \(\mathrm{x}(2\mathrm{x} + 7) = 2\mathrm{x} + 7\mathrm{x} = 9\mathrm{x}\) (treating it as addition instead of distribution)

When they compare this incorrect expansion to \(2\mathrm{x}^2 + \mathrm{ax}\), they can't match the terms properly, leading to confusion and guessing.

Second Most Common Error:

Weak INFER skill: Students don't recognize that "equivalent expressions" means equal coefficients

They might expand correctly to get \(2\mathrm{x}^2 + 7\mathrm{x}\) but then not understand how to use this information to find a. They may try substituting specific values of x or use other incorrect approaches instead of directly comparing coefficients.

This leads to confusion and abandoning systematic solution in favor of guessing.

The Bottom Line:

This problem tests whether students can fluently use the distributive property and understand that polynomial equivalence means coefficient equality. The algebraic manipulation is straightforward, but students must execute it precisely and recognize the conceptual relationship between equivalent expressions.