The square of a positive integer exceeds twenty times the integer by 96. What is that positive integer?

Introduction:
This question tests forming a quadratic equation from a word statement and solving it. The phrase “the square of an integer exceeds twenty times the integer by 96” means: integer^2 is 96 more than 20 times the integer. Translating that carefully produces a quadratic equation. Because the integer is stated to be positive, we choose only the positive root after solving.

Given Data / Assumptions:

• Let the positive integer be n
• “Square exceeds 20 times the integer by 96” means n^2 = 20n + 96
• We solve for integer n and keep only n > 0

Concept / Approach:
Convert the statement into a quadratic equation: n^2 - 20n - 96 = 0. Solve using factoring or the quadratic formula. Since the problem requires a positive integer, discard any negative solution even if it satisfies the equation algebraically.

Step-by-Step Solution:
Start with: n^2 = 20n + 96 Bring all terms to one side: n^2 - 20n - 96 = 0 Find factors of -96 whose difference is 20: 24 and -4 So (n - 24)(n + 4) = 0 n = 24 or n = -4 Positive integer solution: n = 24

Verification / Alternative check:
Check n = 24: n^2 = 576 and 20n = 480. Difference = 576 - 480 = 96, which matches the statement exactly. So 24 is correct.

Why Other Options Are Wrong:
36 gives 36^2 - 20*36 = 1296 - 720 = 576, not 96. 42 and 48 produce even larger differences. 16 gives 256 - 320 = -64, not 96.

Common Pitfalls:
Misreading “exceeds by 96” as “is 96 times,” or forgetting to enforce the “positive integer” condition and mistakenly choosing -4.

Final Answer:
The positive integer is 24.