sqrt(4x^2 - 12x + 9) = 6If x gt 0 and x is a solution to the given equation, what...

1. TRANSLATE the problem information

• Given information:
  • \(\sqrt{4\mathrm{x}^2 - 12\mathrm{x} + 9} = 6\)
  • \(\mathrm{x} \gt 0\) (constraint)
• Find: value of \(2\mathrm{x} - 3\)

2. INFER the key insight about the expression under the square root

• Look at \(4\mathrm{x}^2 - 12\mathrm{x} + 9\) and check if it's a perfect square trinomial
• First term: \(4\mathrm{x}^2 = (2\mathrm{x})^2\)
• Last term: \(9 = 3^2\)
• Middle term: \(-12\mathrm{x} = -2(2\mathrm{x})(3)\)
• This matches the pattern \(\mathrm{a}^2 - 2\mathrm{ab} + \mathrm{b}^2 = (\mathrm{a} - \mathrm{b})^2\)
• Therefore: \(4\mathrm{x}^2 - 12\mathrm{x} + 9 = (2\mathrm{x} - 3)^2\)

3. SIMPLIFY the equation using the perfect square form

• Substitute: \(\sqrt{(2\mathrm{x} - 3)^2} = 6\)
• Apply the square root property: \(\sqrt{\mathrm{a}^2} = |\mathrm{a}|\)
• This gives us: \(|2\mathrm{x} - 3| = 6\)

4. CONSIDER ALL CASES for the absolute value equation

• \(|2\mathrm{x} - 3| = 6\) means either:
  • Case 1: \(2\mathrm{x} - 3 = 6\)
  • Case 2: \(2\mathrm{x} - 3 = -6\)

• Solve Case 1: \(2\mathrm{x} - 3 = 6\)
  • \(2\mathrm{x} = 9\)
  • \(\mathrm{x} = \frac{9}{2} = 4.5\)

• Solve Case 2: \(2\mathrm{x} - 3 = -6\)
  • \(2\mathrm{x} = -3\)
  • \(\mathrm{x} = -\frac{3}{2} = -1.5\)

5. APPLY CONSTRAINTS to select the valid solution

• Check which solutions satisfy \(\mathrm{x} \gt 0\):
  • \(\mathrm{x} = 4.5 \gt 0\) ✓ (valid)
  • \(\mathrm{x} = -1.5 \lt 0\) ✗ (invalid)
• Therefore \(\mathrm{x} = \frac{9}{2}\) is our solution

6. Calculate the final answer

• With \(\mathrm{x} = \frac{9}{2}\):

\(2\mathrm{x} - 3 = 2(\frac{9}{2}) - 3\)
\(= 9 - 3\)
\(= 6\)

Answer: D. 6

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that \(4\mathrm{x}^2 - 12\mathrm{x} + 9\) is a perfect square trinomial

Students often try to solve \(\sqrt{4\mathrm{x}^2 - 12\mathrm{x} + 9} = 6\) by squaring both sides immediately, getting \(4\mathrm{x}^2 - 12\mathrm{x} + 9 = 36\), then \(4\mathrm{x}^2 - 12\mathrm{x} - 27 = 0\). This leads to a messy quadratic that's difficult to solve and doesn't leverage the elegant structure of the problem. This approach often leads to calculation errors and confusion, causing them to abandon systematic solution and guess.

Second Most Common Error:

Incomplete CONSIDER ALL CASES execution: Forgetting the absolute value creates two cases

Some students correctly identify the perfect square trinomial and get \(|2\mathrm{x} - 3| = 6\), but then only solve \(2\mathrm{x} - 3 = 6\), missing the negative case entirely. This gives them \(\mathrm{x} = 4.5\) and \(2\mathrm{x} - 3 = 6\), which happens to be correct, but they miss the complete mathematical reasoning and might not recognize why their constraint \(\mathrm{x} \gt 0\) matters.

The Bottom Line:

The key insight that transforms this problem from difficult to straightforward is recognizing the perfect square trinomial pattern. Without this insight, students face unnecessary algebraic complexity that often leads to errors.