\(\mathrm{x + 23 = (x + 23)(x - c)}\)Which of the following are solutions to the given equation, where c...

1. SIMPLIFY the equation to standard form

• Start with: \(\mathrm{x + 23 = (x + 23)(x - c)}\)
• Move everything to one side: \(\mathrm{x + 23 - (x + 23)(x - c) = 0}\)
• Factor out the common term \(\mathrm{(x + 23)}\): \(\mathrm{(x + 23)[1 - (x - c)] = 0}\)
• Clean up the second bracket: \(\mathrm{(x + 23)[c + 1 - x] = 0}\)

2. INFER the solution strategy

• We now have a product equal to zero, so we can use the zero product property
• Either \(\mathrm{x + 23 = 0}\) OR \(\mathrm{c + 1 - x = 0}\)
• This gives us: \(\mathrm{x = -23}\) OR \(\mathrm{x = c + 1}\)

3. APPLY CONSTRAINTS to test each option

Option I: x = c

• Substitute into original: \(\mathrm{c + 23 = (c + 23)(c - c) = (c + 23)(0) = 0}\)
• This requires \(\mathrm{c + 23 = 0}\), meaning \(\mathrm{c = -23}\)
• But we're told \(\mathrm{c \gt 15}\), so this is impossible ✗

Option II: x = c + 1

• This matches one of our solutions directly ✓

Option III: x = -23

• This matches our other solution directly ✓

Answer: C (II and III only)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skills: Students struggle with factoring out \(\mathrm{(x + 23)}\) from the rearranged equation, getting stuck on the algebraic manipulation. They might try to expand \(\mathrm{(x + 23)(x - c)}\) instead of factoring, leading to a messy quadratic that's harder to solve.

This leads to confusion and abandoning systematic solution, causing them to guess.

Second Most Common Error:

Missing APPLY CONSTRAINTS reasoning: Students correctly find that the equation yields \(\mathrm{x = -23}\) and \(\mathrm{x = c + 1}\), but when testing option I \(\mathrm{(x = c)}\), they don't recognize that this would require \(\mathrm{c = -23}\), which violates the given constraint \(\mathrm{c \gt 15}\). They incorrectly conclude that \(\mathrm{x = c}\) is also a solution.

This may lead them to select Choice D (I, II and III).

The Bottom Line:

This problem requires solid factoring skills combined with careful constraint checking. The algebraic manipulation isn't obvious, and students must remember to verify that their solutions are consistent with the given conditions.