\((\mathrm{x} - 47)^2 = 1\) What is the sum of the solutions to the given equation?...

1. CONSIDER ALL CASES when taking square roots

• Given equation: \((x - 47)^2 = 1\)
• When we take the square root of both sides, we must consider both positive and negative cases:
  • \(x - 47 = +1\) AND \(x - 47 = -1\)

2. SIMPLIFY each case separately

• Case 1: \(x - 47 = 1\)
  • Add 47 to both sides: \(x = 1 + 47 = 48\)
• Case 2: \(x - 47 = -1\)
  • Add 47 to both sides: \(x = -1 + 47 = 46\)

3. Find the sum of all solutions

• We found \(x = 48\) and \(x = 46\)
• \(\mathrm{Sum} = 48 + 46 = 94\)

Answer: 94

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 case, solving \(x - 47 = 1\) to get \(x = 48\), then incorrectly answering 48 instead of recognizing there are two solutions that need to be summed.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify both cases but make arithmetic errors when adding 47 (such as \(47 + 1 = 47\) or \(47 - 1 = 47\)), leading to incorrect individual solutions and therefore an incorrect sum.

The Bottom Line:

The key challenge is remembering that equations of the form \(a^2 = k\) always have two solutions when \(k \gt 0\), and problems asking for the "sum of solutions" require finding ALL solutions first.