\(\mathrm{x}^2 = (22)(22)\) What is the positive solution to the given equation?...

1. TRANSLATE the equation structure

• Given equation: \(\mathrm{x^2 = (22)(22)}\)
• Recognize that \(\mathrm{(22)(22)}\) means \(\mathrm{22 \times 22}\), which equals \(\mathrm{22^2}\)
• Equation becomes: \(\mathrm{x^2 = 22^2}\)

2. INFER the solution strategy

• When you have \(\mathrm{x^2 = (\text{some number})^2}\), take the square root of both sides
• This is the most direct path to isolate x

3. CONSIDER ALL CASES when taking square roots

• Taking square root of both sides: \(\mathrm{\sqrt{x^2} = \sqrt{22^2}}\)
• This gives us: \(\mathrm{x = \pm 22}\)
• Remember: square roots always give both positive and negative solutions

4. APPLY CONSTRAINTS to select the final answer

• The problem asks specifically for the "positive solution"
• From \(\mathrm{x = \pm 22}\), select only the positive value
• Therefore: \(\mathrm{x = 22}\)

Answer: 22

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students don't recognize that \(\mathrm{(22)(22) = 22^2}\) and instead compute \(\mathrm{22 \times 22 = 484}\), then solve \(\mathrm{x^2 = 484}\).

While this approach still leads to the correct answer (\(\mathrm{x = \pm 22}\)), it involves unnecessary arithmetic that creates opportunities for calculation errors. Some students might compute 484 incorrectly, or make errors when finding \(\mathrm{\sqrt{484}}\). This computational detour can lead to confusion and wrong answer selection.

Second Most Common Error:

Inadequate CONSIDER ALL CASES execution: Students take the square root but forget that it yields both positive and negative solutions.

They might write \(\mathrm{x = 22}\) directly from \(\mathrm{\sqrt{x^2} = \sqrt{22^2}}\), missing the ± entirely. While this happens to give the correct final answer since we want the positive solution anyway, it demonstrates incomplete understanding of square root operations that will hurt them in other problems.

The Bottom Line:

This problem rewards pattern recognition over computation. Students who immediately see \(\mathrm{(22)(22)}\) as \(\mathrm{22^2}\) can solve it in seconds, while those who don't recognize this pattern get bogged down in arithmetic and create unnecessary opportunities for error.