What value of x satisfies the equation sqrt(x + 9) = sqrt(3x + 1) - 2?(Grid-in response)

1. APPLY CONSTRAINTS to establish the valid domain

• Given equation: \(\sqrt{\mathrm{x + 9}} = \sqrt{\mathrm{3x + 1}} - 2\)
• Domain requirements:
  • \(\mathrm{x + 9} \geq 0\), so \(\mathrm{x} \geq -9\)
  • \(\mathrm{3x + 1} \geq 0\), so \(\mathrm{x} \geq -\frac{1}{3}\)
  • \(\sqrt{\mathrm{3x + 1}} - 2 \geq 0\), so \(\sqrt{\mathrm{3x + 1}} \geq 2\), meaning \(\mathrm{3x + 1} \geq 4\), so \(\mathrm{x} \geq 1\)
• Therefore: \(\mathrm{x} \geq 1\)

2. INFER the solving strategy

• We have radicals on both sides, so we'll need to square
• But squaring \(\sqrt{\mathrm{x + 9}} = \sqrt{\mathrm{3x + 1}} - 2\) directly would create a messy expansion
• Better strategy: rearrange first to isolate one radical, then square

3. SIMPLIFY by rearranging and squaring

• Rearrange: \(\sqrt{\mathrm{3x + 1}} = \sqrt{\mathrm{x + 9}} + 2\)
• Square both sides: \(\mathrm{3x + 1} = (\sqrt{\mathrm{x + 9}} + 2)^2\)
• Expand the right side: \(\mathrm{3x + 1} = (\mathrm{x + 9}) + 4\sqrt{\mathrm{x + 9}} + 4\)
• Combine: \(\mathrm{3x + 1} = \mathrm{x + 13} + 4\sqrt{\mathrm{x + 9}}\)

4. SIMPLIFY further to isolate the remaining radical

• Collect terms: \(\mathrm{3x + 1 - x - 13} = 4\sqrt{\mathrm{x + 9}}\)
• Simplify: \(\mathrm{2x - 12} = 4\sqrt{\mathrm{x + 9}}\)
• Divide by 2: \(\mathrm{x - 6} = 2\sqrt{\mathrm{x + 9}}\)

5. SIMPLIFY by squaring again

• Square both sides: \((\mathrm{x - 6})^2 = 4(\mathrm{x + 9})\)
• Expand: \(\mathrm{x}^2 - 12\mathrm{x} + 36 = 4\mathrm{x} + 36\)
• Rearrange: \(\mathrm{x}^2 - 12\mathrm{x} + 36 - 4\mathrm{x} - 36 = 0\)
• Simplify: \(\mathrm{x}^2 - 16\mathrm{x} = 0\)
• Factor: \(\mathrm{x}(\mathrm{x - 16}) = 0\)
• Solutions: \(\mathrm{x} = 0\) or \(\mathrm{x} = 16\)

6. APPLY CONSTRAINTS to check validity

• \(\mathrm{x} = 0\): Fails our domain requirement \(\mathrm{x} \geq 1\)
• \(\mathrm{x} = 16\): Satisfies \(\mathrm{x} \geq 1\), so it's potentially valid

7. APPLY CONSTRAINTS by checking the original equation

• For \(\mathrm{x} = 16\):
  • Left side: \(\sqrt{\mathrm{16 + 9}} = \sqrt{25} = 5\)
  • Right side: \(\sqrt{\mathrm{3(16) + 1}} - 2 = \sqrt{49} - 2 = 7 - 2 = 5\) ✓

Answer: 16

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak APPLY CONSTRAINTS skill: Students fail to establish proper domain restrictions at the start or don't check their final answers against both the domain and original equation.

Many students solve the quadratic and get \(\mathrm{x} = 0\) and \(\mathrm{x} = 16\), then conclude the answer is "0 or 16" without realizing that \(\mathrm{x} = 0\) doesn't satisfy the domain requirements. Since this is a grid-in response requiring a single numerical answer, they might enter 0 (the smaller value) instead of the correct answer 16.

Second Most Common Error:

Poor INFER skill about squaring strategy: Students square the original equation \(\sqrt{\mathrm{x + 9}} = \sqrt{\mathrm{3x + 1}} - 2\) directly without rearranging first.

This creates: \(\mathrm{x + 9} = (\sqrt{\mathrm{3x + 1}} - 2)^2 = (\mathrm{3x + 1}) - 4\sqrt{\mathrm{3x + 1}} + 4\), leading to a more complex expression with the radical term having a negative coefficient. The additional algebraic complexity increases chances for arithmetic errors and makes the problem much harder to solve systematically.

The Bottom Line:

This problem tests whether students can manage domain restrictions throughout the entire solution process, not just when solving. Many students can handle the algebra but lose points by including invalid solutions or making the problem unnecessarily complex through poor strategic choices.