No-Calculator Section(x^4-a^4)/(sqrt(x^2+a^2)) = 12sqrt(x^2-a^2)In the given equation, a is a positive constant. Which of the following is one of the...

1. TRANSLATE the problem information

• Given equation: \(\frac{x^4-a^4}{\sqrt{x^2+a^2}} = 12\sqrt{x^2-a^2}\)
• We need to find possible values of \(x\)
• Note: For this to be defined, we need \(x^2-a^2 \gt 0\), so \(|x| \gt a\)

2. INFER the factorization strategy

• The numerator \(x^4-a^4\) looks like a difference of squares pattern
• We can write this as \((x^2)^2 - (a^2)^2\) and factor it
• This should help simplify the complex fraction

3. SIMPLIFY using difference of squares

• Factor: \(x^4-a^4 = (x^2-a^2)(x^2+a^2)\)
• Substitute into the equation:
  \(\frac{(x^2-a^2)(x^2+a^2)}{\sqrt{x^2+a^2}} = 12\sqrt{x^2-a^2}\)

4. SIMPLIFY the fraction

• Since \(x^2+a^2 \gt 0\), we can write: \(\frac{x^2+a^2}{\sqrt{x^2+a^2}} = \sqrt{x^2+a^2}\)
• This gives us: \((x^2-a^2)\sqrt{x^2+a^2} = 12\sqrt{x^2-a^2}\)

5. INFER the next algebraic step

• Since \(x^2-a^2 \gt 0\) from our domain restriction, we can divide both sides by \(\sqrt{x^2-a^2}\)
• This isolates the remaining radical terms: \(\sqrt{x^2-a^2} \cdot \sqrt{x^2+a^2} = 12\)

6. SIMPLIFY using radical properties

• Apply \(\sqrt{A}\sqrt{B} = \sqrt{AB}\): \(\sqrt{(x^2-a^2)(x^2+a^2)} = 12\)
• Expand: \(\sqrt{x^4-a^4} = 12\)
• Square both sides: \(x^4-a^4 = 144\)
• Solve: \(x^4 = a^4 + 144\)

7. CONSIDER ALL CASES when taking roots

• Take the fourth root: \(x = \pm \sqrt[4]{a^4 + 144}\)
• Both solutions satisfy our domain constraint \(|x| \gt a\) since \(\sqrt[4]{a^4 + 144} \gt a\)

Answer: B. \(-\sqrt[4]{a^4 + 144}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students struggle with the initial factorization of \(x^4-a^4\) or get confused by the complex fraction with radicals in both numerator and denominator.

Many students see \(x^4-a^4\) and don't immediately recognize it as a difference of squares that can be factored. They might try to work with it directly or attempt to take fourth roots too early, leading to messy algebra that doesn't simplify nicely. This causes them to get stuck and abandon systematic solution, often guessing among the answer choices.

Second Most Common Error:

Poor CONSIDER ALL CASES execution: Students correctly solve to get \(x^4 = a^4 + 144\) but forget that taking the fourth root of both sides yields both positive and negative solutions.

They might only consider \(x = \sqrt[4]{a^4 + 144}\) and not realize that \(x = -\sqrt[4]{a^4 + 144}\) is also valid. Since the question asks for "one of the possible values" and only provides negative options, this oversight could lead them to think they made an error and guess randomly.

The Bottom Line:

This problem tests both sophisticated algebraic manipulation skills and careful attention to multiple solutions when dealing with even roots. Success requires systematically factoring complex expressions and remembering that even roots always produce both positive and negative solutions.