x^2/25 = 36What is a solution to the given equation?

1. TRANSLATE the problem information

• Given equation: \(\mathrm{x^2/25 = 36}\)
• Find: A solution to this equation from the given choices

2. INFER the solving strategy

• Since \(\mathrm{x^2}\) appears in a fraction, we need to isolate \(\mathrm{x^2}\) first
• Then we can take the square root to find x
• Strategy: Clear the fraction by multiplying both sides by 25

3. SIMPLIFY by clearing the fraction

• Multiply both sides by 25:
  \(\mathrm{x^2/25 \times 25 = 36 \times 25}\)
• Left side simplifies: \(\mathrm{x^2 = 36 \times 25}\)
• Calculate right side: \(\mathrm{x^2 = 900}\)

4. SIMPLIFY by taking the square root

• Take square root of both sides: \(\mathrm{\sqrt{x^2} = \sqrt{900}}\)
• This gives us: \(\mathrm{x = \pm 30}\)

5. INFER which solution fits the answer choices

• We found \(\mathrm{x = 30}\) or \(\mathrm{x = -30}\)
• Looking at choices: A. 6, B. 30, C. 450, D. 900
• Only \(\mathrm{x = 30}\) appears as an option

Answer: B. 30

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when calculating \(\mathrm{36 \times 25}\) or when finding \(\mathrm{\sqrt{900}}\).

For example, they might incorrectly calculate \(\mathrm{36 \times 25 = 180}\) (confusing this with \(\mathrm{36 \times 5}\)), leading them to think \(\mathrm{x^2 = 180}\), so \(\mathrm{x \approx 13.4}\). This doesn't match any answer choice, causing confusion and guessing.

Second Most Common Error:

Poor INFER reasoning: Students try to "distribute" the square operation incorrectly, thinking \(\mathrm{x^2/25 = 36}\) means \(\mathrm{x/5 = 6}\), so \(\mathrm{x = 30}\).

While this accidentally gives the right answer, the reasoning is mathematically incorrect and would fail on similar problems. Students using this flawed approach might get lucky here but struggle with variations.

The Bottom Line:

This problem tests whether students can systematically clear fractions and handle square root operations. The key is recognizing that you must isolate \(\mathrm{x^2}\) completely before taking the square root, rather than trying shortcuts that bypass proper algebraic steps.