The function h is defined by \(\mathrm{h(x) = \sqrt{x - 3} + 2}\). For which value of x is \(\mathrm{h(x)...

1. TRANSLATE the problem information

• Given: \(\mathrm{h(x) = \sqrt{x - 3} + 2}\)
• Find: the value of x when \(\mathrm{h(x) = 7}\)
• This means we need: \(\mathrm{\sqrt{x - 3} + 2 = 7}\)

2. INFER the solving strategy

• We have a square root equation to solve
• Key strategy: isolate the radical term first, then square both sides
• This prevents complications from squaring the entire left side

3. SIMPLIFY by isolating the radical

• Start with: \(\mathrm{\sqrt{x - 3} + 2 = 7}\)
• Subtract 2 from both sides: \(\mathrm{\sqrt{x - 3} = 5}\)

4. SIMPLIFY by eliminating the square root

• Square both sides: \(\mathrm{(\sqrt{x - 3})^2 = 5^2}\)
• This gives us: \(\mathrm{x - 3 = 25}\)

5. SIMPLIFY to find the final answer

• Add 3 to both sides: \(\mathrm{x = 28}\)
• Let's verify: \(\mathrm{h(28) = \sqrt{28 - 3} + 2 = \sqrt{25} + 2 = 5 + 2 = 7}\) ✓

Answer: C (28)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students correctly set up \(\mathrm{\sqrt{x - 3} + 2 = 7}\) and isolate to get \(\mathrm{\sqrt{x - 3} = 5}\), then square both sides to get \(\mathrm{x - 3 = 25}\), but forget the final step of adding 3 to both sides.

They stop at \(\mathrm{x - 3 = 25}\) and mistakenly think \(\mathrm{x = 25}\), leading them to select Choice A (25).

Second Most Common Error:

Poor INFER reasoning about squaring strategy: Students attempt to square both sides immediately without isolating the radical first, leading to the complicated equation \(\mathrm{(\sqrt{x - 3} + 2)^2 = 49}\). When they try to expand the left side, they make algebraic errors or get confused about how to handle the mixed terms.

This leads to confusion and incorrect calculations, causing them to abandon systematic solution and guess among the remaining choices.

The Bottom Line:

This problem tests whether students can systematically work with radical equations by following the correct sequence: isolate the radical, then eliminate it by squaring. Success depends on both strategic thinking and careful algebraic execution through multiple steps.