\((3\mathrm{x} - 4)^2 = 49\) What is the sum of the solutions to the given equation?...

1. TRANSLATE the problem information

• Given equation: \((3\mathrm{x} - 4)^2 = 49\)
• Find: The sum of all solutions

2. CONSIDER ALL CASES when taking square roots

• Taking the square root of both sides: \(\sqrt{(3\mathrm{x} - 4)^2} = \sqrt{49}\)
• This simplifies to: \(|3\mathrm{x} - 4| = 7\)
• The absolute value equation gives us two cases:
  • Case 1: \(3\mathrm{x} - 4 = 7\)
  • Case 2: \(3\mathrm{x} - 4 = -7\)

3. SIMPLIFY each case separately

Case 1: \(3\mathrm{x} - 4 = 7\)

• Add 4 to both sides: \(3\mathrm{x} = 11\)
• Divide by 3: \(\mathrm{x} = \frac{11}{3}\)

Case 2: \(3\mathrm{x} - 4 = -7\)

• Add 4 to both sides: \(3\mathrm{x} = -3\)
• Divide by 3: \(\mathrm{x} = -1\)

4. SIMPLIFY to find the sum

• Sum = \(\frac{11}{3} + (-1)\)
• Convert to common denominator: Sum = \(\frac{11}{3} - \frac{3}{3}\)
• Sum = \(\frac{11 - 3}{3} = \frac{8}{3}\)

Answer: (C) \(\frac{8}{3}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak CONSIDER ALL CASES skill: Students take the square root but only consider the positive solution.

When they see \(\sqrt{49} = 7\), they only solve \(3\mathrm{x} - 4 = 7\), getting \(\mathrm{x} = \frac{11}{3}\). They miss the negative case entirely and might select Choice (B) \(\frac{14}{3}\) (which would be \(2 \times \frac{11}{3}\)) or get confused trying to match their single solution to the answer choices.

Second Most Common Error:

Poor TRANSLATE reasoning: Students solve correctly for both values (\(\frac{11}{3}\) and \(-1\)) but misunderstand what the question is asking.

They might report just one solution or try to find the product instead of the sum, leading to confusion and guessing among the answer choices.

The Bottom Line:

This problem tests whether students understand that squared expressions create two possible solutions. The key insight is recognizing that \((3\mathrm{x} - 4)^2 = 49\) means the expression inside the parentheses can equal either \(+7\) or \(-7\).