k^2 - 53 = 91 What is the positive solution to the given equation?...

1. SIMPLIFY the equation to isolate k²

• Given: \(\mathrm{k^2 - 53 = 91}\)
• Add 53 to both sides: \(\mathrm{k^2 = 91 + 53 = 144}\)

2. SIMPLIFY further by taking the square root

• Take the square root of both sides: \(\mathrm{k = \sqrt{144}}\)

3. CONSIDER ALL CASES for the square root

• Remember that \(\mathrm{\sqrt{144} = \pm 12}\)
• This gives us two solutions: \(\mathrm{k = 12}\) and \(\mathrm{k = -12}\)

4. APPLY CONSTRAINTS to select the final answer

• The problem asks for "the positive solution"
• Therefore: \(\mathrm{k = 12}\)

Answer: D. 12

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about variables vs. their squares: Some students see \(\mathrm{k^2 = 144}\) and incorrectly conclude that \(\mathrm{k = 144}\), forgetting that \(\mathrm{k^2}\) means "k squared," not just k.

This leads them to think the answer is 144 and select Choice A (144).

The Bottom Line:

This problem tests whether students can systematically work through a simple quadratic equation and remember that square roots produce both positive and negative solutions. The key challenge is maintaining precision in algebraic manipulation and not confusing a variable with its square.