Twenty times a positive integer is less than its square by 96. What is the…

Concept: A word problem translates into an equation phrase by phrase. "A is less than B by C" means B − A = C. Let the positive integer be x; "its square" is x2 and "twenty times" the integer is 20x, so the stated relation becomes the quadratic x2 − 20x = 96.

1. Set up the equation: x2 − 20x = 96, i.e. x2 − 20x − 96 = 0.

2. Factor the quadratic: find two numbers whose product is −96 and whose sum is −20 — these are −24 and 4, since (−24) × 4 = −96 and (−24) + 4 = −20.

3. So x2 − 20x − 96 = (x − 24)(x + 4) = 0, giving x = 24 or x = −4.

4. The problem restricts the integer to be positive, so reject x = −4 and keep x = 24.

Cross-check: 242 − 20 × 24 = 576 − 480 = 96, exactly the difference stated in the question, confirming the value.