The expression \(3(\mathrm{x} - 2)^2 - 27\) can be written in the form \(3(\mathrm{x} - (2 - \mathrm{k}))(\mathrm{x} - (2...

1. TRANSLATE the problem information

• Given: Expression \(\mathrm{3(x - 2)^2 - 27}\)
• Target form: \(\mathrm{3(x - (2 - k))(x - (2 + k))}\)
• Find: The value of k

2. INFER the solution strategy

• The target form suggests the expression should factor into two binomial terms
• The structure looks like it could involve a difference of squares pattern
• I need to manipulate the given expression to match the target form

3. SIMPLIFY by factoring out the common factor

Factor out 3 from both terms:

\(\mathrm{3(x - 2)^2 - 27 = 3[(x - 2)^2 - 9]}\)

4. INFER the difference of squares pattern

• Notice that \(\mathrm{9 = 3^2}\), so the expression becomes: \(\mathrm{3[(x - 2)^2 - 3^2]}\)
• This is a difference of squares: \(\mathrm{a^2 - b^2 = (a - b)(a + b)}\)
• Here, \(\mathrm{a = (x - 2)}\) and \(\mathrm{b = 3}\)

5. SIMPLIFY using the difference of squares formula

\(\mathrm{3[(x - 2)^2 - 3^2]}\)

\(\mathrm{= 3[(x - 2 - 3)(x - 2 + 3)]}\)

\(\mathrm{= 3[(x - 5)(x + 1)]}\)

6. TRANSLATE by matching to the target form

• Current form: \(\mathrm{3(x - 5)(x + 1)}\)
• Target form: \(\mathrm{3(x - (2 - k))(x - (2 + k))}\)
• Match the factors:
  • \(\mathrm{x - 5 = x - (2 + k)}\) → \(\mathrm{2 + k = 5}\) → \(\mathrm{k = 3}\)
  • \(\mathrm{x + 1 = x - (2 - k)}\) → \(\mathrm{2 - k = -1}\) → \(\mathrm{k = 3}\)

Answer: C (3)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing the difference of squares pattern after factoring out 3

Students often get stuck after writing \(\mathrm{3[(x - 2)^2 - 9]}\) and don't see that \(\mathrm{9 = 3^2}\). They might try to expand \(\mathrm{(x - 2)^2}\) first, leading to a messier expression that's harder to factor. Without recognizing the difference of squares, they can't efficiently reach the target form and end up guessing.

Second Most Common Error:

Poor TRANSLATE reasoning: Incorrectly matching the factored form to the target form

After successfully factoring to \(\mathrm{3(x - 5)(x + 1)}\), students sometimes make sign errors when setting up the equations. They might write \(\mathrm{x - 5 = x - (2 - k)}\) instead of \(\mathrm{x - 5 = x - (2 + k)}\), leading to incorrect values of k. This systematic error in matching could lead them to select Choice A (√3) or get confused about the signs.

The Bottom Line:

This problem tests the ability to recognize algebraic patterns and perform systematic form matching. Success requires seeing the difference of squares structure hidden within the expression, then carefully aligning the resulting factors with the target form to solve for the parameter.