Common mistake
Stop after setting a discriminant to zero or solving the squared equation, without checking the radical’s sign and domain restrictions.
Original PassWyze practice
A hard radical equation where algebra alone is not enough
Try the problem first. Then follow the substitution, minimum-value argument, domain check, and verification that make the solution complete.
Mathematics independently checked August 24, 2026
The problem
Find the minimum possible value of 4k
The equation √(k − x) = 64 − x has exactly one real solution when k is as small as possible. What is 4k?
Pause here if you want to solve it independently. The hidden condition comes from the original radical equation, not from the quadratic you get after squaring.
Worked solution
Transform, minimize, and verify
- 1
Expose the hidden condition
Let y = 64 − x. Since a square root cannot be negative, the original equation requires y ≥ 0.
- 2
Transform the equation
Because x = 64 − y, substitution and squaring give k − 64 + y = y², so k = 64 + y² − y.
- 3
Find and check the minimum
Complete the square: k = 255/4 + (y − 1/2)². The minimum occurs at y = 1/2, which satisfies y ≥ 0.
Answer: 4k = 255. At the minimum, k = 255/4 and x = 63.5. In the original equation, the radicand is 0.25, so the left side is 0.5; the right side is also 64 − 63.5 = 0.5.
Reusable lesson
Why the original equation still matters
Squaring can create candidates that satisfy the transformed quadratic but not the original radical equation. A negative value of 64 − x can never equal a principal square root.
Reliable sequence
Transform the equation, solve or minimize, enforce the original sign or domain condition, and substitute the result back into the original equation.
Source posture: This is original PassWyze practice inspired by the skill pattern of College Board Question Bank item 2c288148. It changes the constant and prompt, does not predict future test content, and does not imply College Board affiliation.
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