Forming and Solving a Quadratic from Words If a number is subtracted from the square of its one-half, the result is 48. What is the square root of that number?

Introduction / Context:
This question translates a verbal algebraic relationship into an equation involving a square term. Careful parsing of “square of its one-half” and “subtracted from” ensures the correct order of operations. Solving the resulting quadratic yields the number; then take its square root as requested.

Given Data / Assumptions:

• Let the number be N.
• Square of its one-half is (N/2)^2 = N^2/4.
• N^2/4 − N = 48.
• We need sqrt(N).

Concept / Approach:
Multiply both sides by 4 to clear the denominator, then solve the quadratic by the standard formula or factoring. Consider both roots, but choose the value that makes sense for the context (typically positive for a “number” unless otherwise stated). Finally, compute the square root of N as asked.

Step-by-Step Solution:
Start: N^2/4 − N = 48.Multiply by 4: N^2 − 4N = 192 → N^2 − 4N − 192 = 0.Discriminant: 16 + 768 = 784; sqrt = 28.N = (4 ± 28)/2 → N = 16 or N = −12.Take the sensible positive value: N = 16 → sqrt(N) = 4.

Verification / Alternative check:
Check N = 16: (16/2)^2 − 16 = 8^2 − 16 = 64 − 16 = 48 (works). The negative root is extraneous for a natural-number context.

Why Other Options Are Wrong:
5, 6, 8, 12 are not sqrt(16); they arise from arithmetic slips or misreading the phrase order.

Common Pitfalls:
Interpreting “square of its one-half” as (N^2)/2; subtracting in the wrong order; discarding the quadratic step too soon.

Final Answer:
4