25x^2 - kx + 4In the quadratic expression above, k is a positive constant. If the expression is the square...

1. TRANSLATE the problem information

• Given information:
  • Expression: \(25\mathrm{x}^2 - \mathrm{kx} + 4\)
  • k is a positive constant
  • The expression is the square of a binomial

• What this tells us: We need to find k such that \(25\mathrm{x}^2 - \mathrm{kx} + 4 = (\mathrm{ax} + \mathrm{b})^2\) for some values a and b.

2. INFER the approach

• Since it's the square of a binomial, we can use the perfect square trinomial pattern
• The general form \((\mathrm{ax} + \mathrm{b})^2 = \mathrm{a}^2\mathrm{x}^2 + 2\mathrm{abx} + \mathrm{b}^2\)
• We'll match coefficients to find k

3. SIMPLIFY by expanding and comparing coefficients

• From \((\mathrm{ax} + \mathrm{b})^2 = \mathrm{a}^2\mathrm{x}^2 + 2\mathrm{abx} + \mathrm{b}^2\)
• Comparing with \(25\mathrm{x}^2 - \mathrm{kx} + 4\):
  • First term: \(\mathrm{a}^2 = 25\) → \(\mathrm{a} = \pm5\)
  • Last term: \(\mathrm{b}^2 = 4\) → \(\mathrm{b} = \pm2\)
  • Middle term: \(2\mathrm{ab}\) corresponds to the coefficient of x

4. INFER the correct sign combination

• We need the middle term to be \(-\mathrm{kx}\) (negative)
• If \(\mathrm{a} = 5\) and \(\mathrm{b} = 2\): \((5\mathrm{x} + 2)^2 = 25\mathrm{x}^2 + 20\mathrm{x} + 4\) (wrong sign)
• If \(\mathrm{a} = 5\) and \(\mathrm{b} = -2\): \((5\mathrm{x} - 2)^2 = 25\mathrm{x}^2 - 20\mathrm{x} + 4\) ✓
• This gives us \(\mathrm{k} = 20\)

5. APPLY CONSTRAINTS to verify

• Since k must be positive: \(\mathrm{k} = 20\) ✓
• Verification: \((5\mathrm{x} - 2)^2 = 25\mathrm{x}^2 - 20\mathrm{x} + 4\)

Answer: C) 20

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the connection between "square of a binomial" and perfect square trinomial patterns. They might try to factor \(25\mathrm{x}^2 - \mathrm{kx} + 4\) directly without understanding what makes it a perfect square, leading to random guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \((\mathrm{ax} + \mathrm{b})^2\) but make sign errors when expanding or comparing coefficients. For example, they might get confused about whether the middle term should be \(+2\mathrm{ab}\) or \(-2\mathrm{ab}\), potentially calculating \(\mathrm{k} = -20\) but then selecting Choice C (20) by applying constraints incorrectly, or getting the wrong value entirely.

The Bottom Line:

This problem requires students to bridge abstract algebraic concepts (perfect square trinomials) with concrete coefficient manipulation. The key insight is recognizing that being "the square of a binomial" gives you a specific algebraic form to work with.