Question:If \(7(\mathrm{a} - 3)^2 = 5(\mathrm{a} - 3)^2 + 18\), what is the value of \((\mathrm{a} - 3)^2\)?036918

1. INFER the most efficient approach

• Notice that \((\mathrm{a} - 3)^2\) appears in both terms on the equation
• Key insight: Instead of expanding this complex expression, use substitution to simplify
• Let \(\mathrm{u} = (\mathrm{a} - 3)^2\) to transform this into a basic linear equation

2. TRANSLATE the original equation using substitution

• Original: \(7(\mathrm{a} - 3)^2 = 5(\mathrm{a} - 3)^2 + 18\)
• With \(\mathrm{u} = (\mathrm{a} - 3)^2\): \(7\mathrm{u} = 5\mathrm{u} + 18\)
• Now we have a simple linear equation to solve

3. SIMPLIFY by solving for u

• Start with: \(7\mathrm{u} = 5\mathrm{u} + 18\)
• Subtract 5u from both sides: \(7\mathrm{u} - 5\mathrm{u} = 18\)
• Combine like terms: \(2\mathrm{u} = 18\)
• Divide both sides by 2: \(\mathrm{u} = 9\)

4. TRANSLATE back to the original expression

• Since \(\mathrm{u} = (\mathrm{a} - 3)^2\), we have: \((\mathrm{a} - 3)^2 = 9\)

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the substitution opportunity and instead try to expand \((\mathrm{a} - 3)^2\) first.

They might expand to get \(7(\mathrm{a}^2 - 6\mathrm{a} + 9) = 5(\mathrm{a}^2 - 6\mathrm{a} + 9) + 18\), then distribute to get \(7\mathrm{a}^2 - 42\mathrm{a} + 63 = 5\mathrm{a}^2 - 30\mathrm{a} + 45 + 18\). This creates a much more complex equation that's prone to calculation errors and doesn't directly give them \((\mathrm{a} - 3)^2\). This leads to confusion and often causes students to abandon the systematic approach and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students use the correct substitution strategy but make arithmetic mistakes.

For example, they might incorrectly calculate \(7\mathrm{u} - 5\mathrm{u} = 3\mathrm{u}\) instead of \(2\mathrm{u}\), or divide 18 by the wrong number. If they get \(3\mathrm{u} = 18\), they'd find \(\mathrm{u} = 6\), leading them to select Choice C (6).

The Bottom Line:

The key insight is recognizing that when you see the same complex expression repeated in an equation, substitution can transform a difficult problem into a simple one. Students who miss this strategy often get lost in unnecessary algebraic complexity.