\((\mathrm{x} - 8)^2 = 49\) What is the sum of the solutions to the given equation?...

1. TRANSLATE the problem information

• Given equation: \((x - 8)^2 = 49\)
• Need to find: Sum of all solutions

2. CONSIDER ALL CASES when taking square roots

• When we have an equation of the form \((\mathrm{expression})^2 = \mathrm{number}\), taking square roots gives us both positive and negative possibilities
• \(\sqrt{(x - 8)^2} = ±\sqrt{49}\)
• This means: \(x - 8 = +7\) OR \(x - 8 = -7\)

3. SIMPLIFY each case separately

• Case 1: \(x - 8 = 7\)
  Add 8 to both sides: \(x = 15\)
• Case 2: \(x - 8 = -7\)
  Add 8 to both sides: \(x = 1\)

4. Find the sum of both solutions

• Sum = \(15 + 1 = 16\)

Answer: 16

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak CONSIDER ALL CASES skill: Students only consider the positive square root case.

They might think: "\(\sqrt{49} = 7\), so \(x - 8 = 7\), therefore \(x = 15\)." They stop here and don't realize there's also a negative case where \(x - 8 = -7\), giving \(x = 1\). Since they only find one solution (\(x = 15\)), they might answer 15 instead of the correct sum of 16.

Second Most Common Error:

Conceptual confusion about square root properties: Students might incorrectly think that since we're looking at \((x - 8)^2\), the expression inside must be positive.

This leads them to only consider \(x - 8 = 7\) (thinking the quantity \(x - 8\) must be positive), missing the fact that \((-7)^2 = 49\) just as much as \(7^2 = 49\). This causes them to find only \(x = 15\) and miss the complete solution set.

The Bottom Line:

The key insight is remembering that when you take the square root of both sides of an equation, you must consider both the positive and negative possibilities. Many students have been trained to think "\(\sqrt{49} = 7\)" without remembering that in equation-solving contexts, we need "\(±7\)" to capture all solutions.