\((\frac{1}{4}\mathrm{x} + \mathrm{k})^2 = \frac{1}{16}\mathrm{x}^2 + \frac{3}{8}\mathrm{x} + \mathrm{k}^2\) If the equation above represents an iden...

1. TRANSLATE the problem information

• Given: \(\left(\frac{1}{4}x + k\right)^2 = \frac{1}{16}x^2 + \frac{3}{8}x + k^2\)
• This is an identity true for all values of x
• We need to find the positive constant k
• What this tells us: The two sides must be exactly the same polynomial

2. INFER the solution approach

• Since we have a perfect square on the left and an expanded form on the right, we should expand the left side
• For an identity, the expanded forms must have identical coefficients for each power of x
• This will give us an equation to solve for k

3. SIMPLIFY by expanding the perfect square

• Use the formula \((a + b)^2 = a^2 + 2ab + b^2\) where \(a = \frac{1}{4}x\) and \(b = k\)
• \(a^2 = \left(\frac{1}{4}x\right)^2 = \frac{1}{16}x^2\)
• \(2ab = 2 \times \frac{1}{4}x \times k = \frac{1}{2}kx\)
• \(b^2 = k^2\)
• So: \(\left(\frac{1}{4}x + k\right)^2 = \frac{1}{16}x^2 + \frac{1}{2}kx + k^2\)

4. INFER the coefficient matching requirement

• Since both sides are identical: \(\frac{1}{16}x^2 + \frac{1}{2}kx + k^2 = \frac{1}{16}x^2 + \frac{3}{8}x + k^2\)
• The coefficients of like terms must be equal:
  • \(x^2\) terms: \(\frac{1}{16} = \frac{1}{16}\) ✓
  • \(x\) terms: \(\frac{1}{2}k = \frac{3}{8}\)
  • Constant terms: \(k^2 = k^2\) ✓

5. SIMPLIFY to solve for k

• From \(\frac{1}{2}k = \frac{3}{8}\)
• Multiply both sides by 2: \(k = 2 \times \frac{3}{8} = \frac{6}{8} = \frac{3}{4}\)

Answer: \(\frac{3}{4}\) (or 0.75)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not understand what "identity that is true for all values of x" means. They might try to solve by substituting specific values of x instead of recognizing that the polynomials must be identical.

This leads to confusion and random guessing rather than systematic expansion and coefficient matching.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly expand the left side but make arithmetic errors in the perfect square expansion, particularly with the middle term \(2ab = 2 \times \frac{1}{4}x \times k\).

Common mistakes include getting \(\frac{k}{4}x\) instead of \(\frac{1}{2}kx\) for the middle term, which leads to the wrong equation and an incorrect value of k.

The Bottom Line:

This problem tests whether students understand polynomial identities conceptually and can execute perfect square expansions accurately. The key insight is recognizing that "true for all x" means the polynomials are identical, not just equal for specific values.