If \((\mathrm{x}+3)^2 = 4\mathrm{x} + 24\), what is the positive value of x?

1. TRANSLATE the problem information

• Given equation: \((\mathrm{x}+3)^2 = 4\mathrm{x} + 24\)
• Need to find: the positive value of x

2. SIMPLIFY by expanding the left side

• Expand \((\mathrm{x}+3)^2\): \(\mathrm{x}^2 + 6\mathrm{x} + 9\)
• Now we have: \(\mathrm{x}^2 + 6\mathrm{x} + 9 = 4\mathrm{x} + 24\)

3. SIMPLIFY to get standard quadratic form

• Move all terms to the left side: \(\mathrm{x}^2 + 6\mathrm{x} + 9 - 4\mathrm{x} - 24 = 0\)
• Combine like terms: \(\mathrm{x}^2 + 2\mathrm{x} - 15 = 0\)

4. SIMPLIFY by factoring the quadratic

• Need two numbers that multiply to -15 and add to 2
• Those numbers are 5 and -3
• Factor: \((\mathrm{x} + 5)(\mathrm{x} - 3) = 0\)

5. APPLY CONSTRAINTS to select the final answer

• Using zero product property: \(\mathrm{x} + 5 = 0\) or \(\mathrm{x} - 3 = 0\)
• This gives us: \(\mathrm{x} = -5\) or \(\mathrm{x} = 3\)
• Since we need the positive value: \(\mathrm{x} = 3\)

Answer: C) 3

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make algebraic errors when expanding \((\mathrm{x}+3)^2\) or when combining like terms after moving everything to one side.

For example, they might expand \((\mathrm{x}+3)^2\) as \(\mathrm{x}^2 + 9\) (forgetting the middle term) or make sign errors when rearranging. This leads to an incorrect quadratic equation that factors differently, causing them to select wrong answer choices or get confused and guess.

Second Most Common Error:

Poor APPLY CONSTRAINTS reasoning: Students correctly solve the quadratic and find both \(\mathrm{x} = -5\) and \(\mathrm{x} = 3\), but then select the negative solution because they don't carefully read that the problem asks for the positive value.

This may lead them to select Choice A (-5).

The Bottom Line:

This problem tests both algebraic manipulation skills and careful reading. Success requires systematic expansion, accurate combining of terms, proper factoring technique, and attention to the constraint that eliminates one of the two mathematically valid solutions.