The equation sqrt(5 - x) + 3 = 0 is given. How many distinct real solutions does the equation have?

1. TRANSLATE the problem information

• Given equation: \(\sqrt{5 - \mathrm{x}} + 3 = 0\)
• Need to find: Number of distinct real solutions

2. INFER the key mathematical constraint

• The symbol √ represents the principal square root
• By definition, principal square roots are always non-negative: \(\sqrt{\text{anything}} \geq 0\)
• This means \(\sqrt{5 - \mathrm{x}} \geq 0\), regardless of what x equals

3. APPLY CONSTRAINTS to analyze the equation

• Since \(\sqrt{5 - \mathrm{x}} \geq 0\), we have:
• \(\sqrt{5 - \mathrm{x}} + 3 \geq 0 + 3 = 3\)
• The left side of our equation is always at least 3
• But we need it to equal 0, which is impossible

4. INFER the final conclusion

• Since \(\sqrt{5 - \mathrm{x}} + 3 \geq 3\) for all real x, and \(3 \gt 0\), the equation has no solutions

Answer: A (0 solutions)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that principal square roots are always non-negative, so they try to solve \(\sqrt{5 - \mathrm{x}} = -3\) algebraically.

They square both sides: \((\sqrt{5 - \mathrm{x}})^2 = (-3)^2\)
This gives: \(5 - \mathrm{x} = 9\), so \(\mathrm{x} = -4\)

When they check: \(\sqrt{5 - (-4)} + 3 = \sqrt{9} + 3 = 3 + 3 = 6 \neq 0\)

This creates an extraneous solution that doesn't work, but students might still think there's one solution and select Choice B (1).

Second Most Common Error:

Missing conceptual knowledge: Students might not remember that the √ symbol specifically means the principal (non-negative) square root, thinking it could represent both positive and negative roots.

This conceptual gap leads to confusion about whether -3 is a valid output for \(\sqrt{5 - \mathrm{x}}\), causing them to guess randomly among the choices.

The Bottom Line:

This problem tests whether students truly understand that the radical symbol √ has a restricted range (non-negative outputs only), not just whether they can manipulate radical equations algebraically.