x^2/25 - 2 = 7 What is the positive solution to the given equation?...

1. TRANSLATE the problem information

• Given equation: \(\frac{\mathrm{x}^2}{25} - 2 = 7\)
• Find: the positive solution for x

2. SIMPLIFY by isolating the \(\frac{\mathrm{x}^2}{25}\) term

• Add 2 to both sides: \(\frac{\mathrm{x}^2}{25} - 2 + 2 = 7 + 2\)
• This gives us: \(\frac{\mathrm{x}^2}{25} = 9\)

3. SIMPLIFY by eliminating the fraction

• Multiply both sides by 25: \(25 \times \frac{\mathrm{x}^2}{25} = 25 \times 9\)
• This simplifies to: \(\mathrm{x}^2 = 225\)

4. SIMPLIFY by taking the square root

• Take the square root of both sides: \(\sqrt{\mathrm{x}^2} = \sqrt{225}\)
• This gives us: \(\mathrm{x} = \pm15\)
• Remember: when we take the square root of both sides, we get both positive and negative solutions

5. APPLY CONSTRAINTS to select the final answer

• The problem asks specifically for the positive solution
• From \(\mathrm{x} = \pm15\), we select \(\mathrm{x} = 15\)

Answer: B (15)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic mistakes during the algebraic manipulations.

Common errors include:

• Adding incorrectly: \(7 + 2 = 8\) instead of 9
• Multiplying incorrectly: \(9 \times 25 = 200\) instead of 225
• Taking square root incorrectly: \(\sqrt{225} = 25\) instead of 15

These calculation errors can lead students to select any of the wrong answer choices or cause them to second-guess their work.

Second Most Common Error:

Incomplete SIMPLIFY process: Students stop their solution prematurely at \(\mathrm{x}^2 = 225\).

When students see \(\mathrm{x}^2 = 225\), they might think this is the final answer and select Choice D (225) without recognizing they need to take one more step to find x itself.

The Bottom Line:

This problem tests careful algebraic manipulation through multiple steps. Each step builds on the previous one, so a single arithmetic error can derail the entire solution. The key is working systematically through each algebraic operation while maintaining accuracy.