\((\mathrm{x} - 2)^2 + 7\) Which of the following is equivalent to the expression above?...

1. INFER the approach needed

• We have a perfect square binomial \((\mathrm{x} - 2)^2\) plus a constant 7
• The strategy is to expand the binomial first, then add the constant term

2. SIMPLIFY by expanding the perfect square

• \((\mathrm{x} - 2)^2 = (\mathrm{x} - 2)(\mathrm{x} - 2)\)
• Using FOIL method:
  • First: \(\mathrm{x} \cdot \mathrm{x} = \mathrm{x}^2\)
  • Outer: \(\mathrm{x} \cdot (-2) = -2\mathrm{x}\)
  • Inner: \((-2) \cdot \mathrm{x} = -2\mathrm{x}\)
  • Last: \((-2) \cdot (-2) = +4\)
• Result: \(\mathrm{x}^2 - 2\mathrm{x} - 2\mathrm{x} + 4 = \mathrm{x}^2 - 4\mathrm{x} + 4\)

3. SIMPLIFY by adding the constant and combining terms

• \((\mathrm{x} - 2)^2 + 7 = \mathrm{x}^2 - 4\mathrm{x} + 4 + 7\)
• Combine the constant terms: \(4 + 7 = 11\)
• Final expression: \(\mathrm{x}^2 - 4\mathrm{x} + 11\)

Answer: B (\(\mathrm{x}^2 - 4\mathrm{x} + 11\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when expanding \((\mathrm{x} - 2)^2\), particularly with the middle term. They might write \((\mathrm{x} - 2)^2 = \mathrm{x}^2 + 4\mathrm{x} + 4\) instead of \(\mathrm{x}^2 - 4\mathrm{x} + 4\), confusing the signs from \((\mathrm{x} - 2)\) vs \((\mathrm{x} + 2)\).

This leads them to get \(\mathrm{x}^2 + 4\mathrm{x} + 4 + 7 = \mathrm{x}^2 + 4\mathrm{x} + 11\) and select Choice D (\(\mathrm{x}^2 + 4\mathrm{x} + 11\)).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly expand to \(\mathrm{x}^2 - 4\mathrm{x} + 4\) but make an arithmetic error when adding constants, calculating \(4 + 7 = 3\) instead of 11.

This causes them to select Choice A (\(\mathrm{x}^2 - 4\mathrm{x} + 3\)).

The Bottom Line:

Perfect square binomial problems require careful attention to signs during expansion and accurate arithmetic when combining terms. The negative coefficient in \((\mathrm{x} - 2)\) creates the most confusion, as students often mix up the expansion patterns for \((\mathrm{x} - \mathrm{b})^2\) versus \((\mathrm{x} + \mathrm{b})^2\).