x^2/(sqrt(x^2-c^2)) = c^2/(sqrt(x^2-c^2)) + 39In the given equation, c is a positive constant. Which of the following is one of...

1. INFER domain restrictions first

• Given equation: \(\mathrm{x^2/\sqrt{x^2-c^2} = c^2/\sqrt{x^2-c^2} + 39}\)
• For the equation to be defined, we need \(\mathrm{x^2 - c^2 \gt 0}\)
• This means \(\mathrm{x^2 \gt c^2}\), so our solutions must satisfy this condition

2. SIMPLIFY by isolating the variable terms

• Subtract \(\mathrm{c^2/\sqrt{x^2-c^2}}\) from both sides:

\(\mathrm{x^2/\sqrt{x^2-c^2} - c^2/\sqrt{x^2-c^2} = 39}\)

• Combine the left side over common denominator:

\(\mathrm{(x^2 - c^2)/\sqrt{x^2-c^2} = 39}\)

3. SIMPLIFY by eliminating the square root

• Square both sides to remove the square root:

\(\mathrm{[(x^2 - c^2)/\sqrt{x^2-c^2}]^2 = 39^2}\)

• This gives us:

\(\mathrm{(x^2 - c^2)^2/(x^2-c^2) = 39^2}\)

• Since \(\mathrm{x^2 - c^2 ≠ 0}\) (from our domain restriction), cancel one factor:

\(\mathrm{x^2 - c^2 = 39^2}\)

4. SIMPLIFY to solve for x

• Add \(\mathrm{c^2}\) to both sides:

\(\mathrm{x^2 = c^2 + 39^2}\)

• Take the square root of both sides

5. CONSIDER ALL CASES for the final solutions

• Taking the square root gives us: \(\mathrm{x = ±\sqrt{c^2 + 39^2}}\)
• Both solutions: \(\mathrm{x = \sqrt{c^2 + 39^2}}\) and \(\mathrm{x = -\sqrt{c^2 + 39^2}}\)
• Both satisfy our domain restriction since \(\mathrm{c^2 + 39^2 \gt c^2}\) (because c is positive)

Answer: D. \(\mathrm{-\sqrt{c^2 + 39^2}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students often make algebraic errors when combining fractions or squaring both sides of the equation. They might incorrectly handle the fraction \(\mathrm{(x^2 - c^2)/\sqrt{x^2-c^2}}\) when squaring, getting confused about how the terms interact. This leads to incorrect intermediate equations and wrong final answers, causing them to get stuck and guess among the answer choices.

Second Most Common Error:

Poor CONSIDER ALL CASES reasoning: Students solve correctly up to \(\mathrm{x^2 = c^2 + 39^2}\) but only consider the positive square root solution \(\mathrm{x = \sqrt{c^2 + 39^2}}\). They don't recognize that the negative solution \(\mathrm{x = -\sqrt{c^2 + 39^2}}\) is also valid and matches choice D. This may lead them to incorrectly conclude that none of the answer choices work, causing confusion and random guessing.

The Bottom Line:

This problem tests whether students can systematically work through rational equations involving square roots while tracking both positive and negative solutions. The algebraic manipulation is straightforward once the approach is clear, but students must be careful with fraction operations and remember that square roots yield two solutions.