Let n be a positive integer such that five times n equals three less than twice the square of n (that is, 5n = 2n^2 − 3). What is the value of n?

Introduction / Context:
Here we turn a verbal relation into a quadratic equation in a positive integer. Recognizing and solving the quadratic cleanly, then applying the positivity constraint, identifies the valid integer solution. This is a direct test of forming and solving quadratic equations.

Given Data / Assumptions:

• n is a positive integer.
• Equation: 5n = 2n^2 − 3.
• We need the value of n that satisfies the equation.

Concept / Approach:
Rearrange the equation into standard quadratic form an^2 + bn + c = 0, then factor or use the quadratic formula. Discard any non-integer or negative solution that does not meet the stated condition on n.

Step-by-Step Solution:

Verification / Alternative check:
Check: Left side 5n = 15; right side 2n^2 − 3 = 2 * 9 − 3 = 15. The equality holds.

Why Other Options Are Wrong:
13, 23, and 33 do not satisfy the equation. 6 does not satisfy it either when substituted.

Common Pitfalls:
Sign errors when moving terms, miscomputing the discriminant, or accepting a negative or fractional root despite the integer requirement.

Final Answer:
3