Let u and v be real numbers such that u + w = b and v + w = c....

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{u + w = b}\)
  • \(\mathrm{v + w = c}\)
  • Need to find: \(\mathrm{2(u - v)^2}\)

2. INFER the solution strategy

• Since we have u and v in terms of w, and we need u - v, let's solve for each variable first
• We can isolate u and v from the given equations, then find their difference

3. SIMPLIFY by solving for u and v

• From \(\mathrm{u + w = b}\): \(\mathrm{u = b - w}\)
• From \(\mathrm{v + w = c}\): \(\mathrm{v = c - w}\)

4. SIMPLIFY to find u - v

• \(\mathrm{u - v = (b - w) - (c - w)}\)
• \(\mathrm{u - v = b - w - c + w}\)
• \(\mathrm{u - v = b - c}\)

5. SIMPLIFY the target expression

• \(\mathrm{2(u - v)^2 = 2(b - c)^2}\)
• Expand \(\mathrm{(b - c)^2}\): \(\mathrm{(b - c)^2 = b^2 - 2bc + c^2}\)
• Therefore: \(\mathrm{2(b - c)^2 = 2(b^2 - 2bc + c^2) = 2b^2 - 4bc + 2c^2}\)

Answer: (B) \(\mathrm{2b^2 - 4bc + 2c^2}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students try to work directly with \(\mathrm{2(u - v)^2}\) without first solving for u and v from the given equations.

They might attempt to substitute \(\mathrm{u + w = b}\) and \(\mathrm{v + w = c}\) directly into the expression, leading to complicated expressions like \(\mathrm{2((b - w) - (c - w))^2}\) without recognizing the simplification. This creates unnecessary complexity and often leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly find \(\mathrm{u - v = b - c}\) but make sign errors when expanding \(\mathrm{(b - c)^2}\).

They might incorrectly expand \(\mathrm{(b - c)^2}\) as \(\mathrm{b^2 + 2bc + c^2}\) (using the wrong sign for the middle term) or forget to distribute the 2 properly. This may lead them to select Choice (A) \(\mathrm{(2b^2 + 4bc + 2c^2)}\).

The Bottom Line:

This problem tests whether students can see the strategic insight that solving for individual variables first dramatically simplifies the target expression. The key breakthrough is recognizing that u - v becomes just b - c after substitution.