Question: 4 + sqrt(x + 1) = 9What value of x is the solution to the given equation?4242515

1. TRANSLATE the problem information

• Given: \(4 + \sqrt{\mathrm{x + 1}} = 9\)
• Find: The value of x

2. INFER the solution strategy

• This is a radical equation (contains a square root)
• Key insight: To solve radical equations, we must first isolate the radical term, then eliminate it by squaring both sides
• First step: Get \(\sqrt{\mathrm{x + 1}}\) by itself on one side

3. SIMPLIFY by isolating the radical

• Subtract 4 from both sides:
  \(4 + \sqrt{\mathrm{x + 1}} - 4 = 9 - 4\)
  \(\sqrt{\mathrm{x + 1}} = 5\)

4. SIMPLIFY by eliminating the square root

• Square both sides to remove the radical:
  \((\sqrt{\mathrm{x + 1}})^2 = 5^2\)
  \(\mathrm{x + 1 = 25}\)

5. SIMPLIFY to find x

• Subtract 1 from both sides:
  \(\mathrm{x + 1 - 1 = 25 - 1}\)
  \(\mathrm{x = 24}\)

6. Verify the solution

• Substitute x = 24 back into original equation:
  \(4 + \sqrt{24 + 1} = 4 + \sqrt{25} = 4 + 5 = 9\) ✓

Answer: B (24)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students attempt to square both sides immediately without isolating the radical first.

They might try: \((4 + \sqrt{\mathrm{x + 1}})^2 = 9^2\), which expands to \(16 + 8\sqrt{\mathrm{x + 1}} + (\mathrm{x + 1}) = 81\), creating a much more complex equation that's difficult to solve. This leads to confusion and often results in guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly isolate the radical to get \(\sqrt{\mathrm{x + 1}} = 5\), but make arithmetic errors when squaring or solving the linear equation.

For example, they might incorrectly calculate \(5^2 = 10\) instead of 25, leading to \(\mathrm{x + 1 = 10}\) and \(\mathrm{x = 9}\). Since 9 isn't among the choices, this causes confusion and random answer selection.

The Bottom Line:

This problem tests whether students know the fundamental strategy for radical equations: isolate first, then eliminate. Students who try to take shortcuts or don't follow the systematic approach typically get overwhelmed by unnecessary algebraic complexity.