sqrt(4x^2) = sqrt(9x^2) How many solutions does the given equation have?...

1. TRANSLATE the problem information

• Given equation: \(\sqrt{4x^2} = \sqrt{9x^2}\)
• Need to find: How many solutions this equation has

2. INFER the key strategy

• The key insight is recognizing that \(\sqrt{a^2} = |a|\) for any real number a
• This means we need to convert both sides using absolute value properties
• This will transform the radical equation into a simpler absolute value equation

3. SIMPLIFY both sides using the square root property

• Left side: \(\sqrt{4x^2} = \sqrt{4} \cdot \sqrt{x^2} = 2\sqrt{x^2} = 2|x|\)
• Right side: \(\sqrt{9x^2} = \sqrt{9} \cdot \sqrt{x^2} = 3\sqrt{x^2} = 3|x|\)
• New equation: \(2|x| = 3|x|\)

4. SIMPLIFY algebraically to solve

• Subtract \(2|x|\) from both sides: \(2|x| - 2|x| = 3|x| - 2|x|\)
• This gives us: \(0 = |x|\)
• Since \(|x| = 0\), we must have \(x = 0\)

5. INFER why this is the only solution

• For \(x = 0\): Both sides equal 0, so the equation is satisfied ✓
• For any \(x \neq 0\): We have \(|x| \gt 0\), so \(2|x| = 3|x|\) would require \(2 = 3\) (impossible!)
• Therefore, \(x = 0\) is the unique solution

Answer: B (Exactly one)

Why Students Usually Falter on This Problem

Most Common Error Path:

Missing conceptual knowledge: \(\sqrt{a^2} = |a|\) property
Students might incorrectly simplify \(\sqrt{4x^2} = 2x\) and \(\sqrt{9x^2} = 3x\), ignoring the absolute value requirement. This leads to \(2x = 3x\), which gives \(x = 0\). While this happens to yield the correct answer, the reasoning is flawed and could lead to errors in similar problems with different contexts.

Second Most Common Error:

Weak INFER skills: Students solve \(2|x| = 3|x|\) correctly to get \(x = 0\), but fail to recognize why this is the only solution. They might think there could be additional solutions they haven't found, potentially selecting Choice C (Exactly two) or Choice D (Infinitely many) due to uncertainty.

The Bottom Line:

This problem tests whether students understand the fundamental relationship between square roots and absolute values. The algebraic steps are straightforward once the conversion is made, but missing this key insight makes the problem much more difficult and prone to conceptual errors.