\(\mathrm{x - 29 = (x - a)(x - 29)}\)Which of the following are solutions to the given equation, where a...

1. SIMPLIFY the equation to standard form

• Start with: \(\mathrm{x - 29 = (x - a)(x - 29)}\)
• Move everything to one side: \(\mathrm{x - 29 - (x - a)(x - 29) = 0}\)
• Factor out the common factor \(\mathrm{(x - 29)}\): \(\mathrm{(x - 29)[1 - (x - a)] = 0}\)

2. SIMPLIFY the bracketed expression

• Expand inside the brackets: \(\mathrm{(x - 29)[1 - x + a] = 0}\)
• Rearrange: \(\mathrm{(x - 29)[a + 1 - x] = 0}\)
• Rewrite in standard form: \(\mathrm{(x - 29)(x - (a + 1)) = 0}\)

3. INFER that zero product property applies

• Since we have a product equal to zero, either factor can equal zero
• This gives us: \(\mathrm{x - 29 = 0}\) OR \(\mathrm{x - (a + 1) = 0}\)

4. SIMPLIFY to find the solutions

• From \(\mathrm{x - 29 = 0}\): \(\mathrm{x = 29}\)
• From \(\mathrm{x - (a + 1) = 0}\): \(\mathrm{x = a + 1}\)

5. CONSIDER ALL CASES for the given options

• Check option I (a): This does NOT equal 29 or \(\mathrm{a + 1}\), so it's not a solution
• Check option II \(\mathrm{(a + 1)}\): This DOES equal one of our solutions ✓
• Check option III (29): This DOES equal one of our solutions ✓

Answer: C. II and III only

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students struggle with factoring out the common \(\mathrm{(x - 29)}\) term correctly, getting confused by the algebraic manipulation needed to reach \(\mathrm{(x - 29)(x - (a + 1)) = 0}\).

Many students might try to expand \(\mathrm{(x - a)(x - 29)}\) first, leading to a much more complicated quadratic equation that's harder to solve. Others might make sign errors when factoring or combining terms. This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Inadequate CONSIDER ALL CASES execution: Students find one correct solution (often \(\mathrm{x = 29}\) since it's more obvious) but don't systematically check all three options or don't find the second solution \(\mathrm{x = a + 1}\).

They might see that 29 works and immediately select an answer containing III, potentially choosing Choice B (I and III only) if they mistakenly think a is also a solution, or get confused about which combinations include 29.

The Bottom Line:

This problem tests whether students can handle algebraic factoring when there's a repeated term and systematically find all solutions. The key insight is recognizing that factoring out the common \(\mathrm{(x - 29)}\) immediately simplifies the problem rather than expanding everything first.