Product and sum of two numbers (quadratic recovery): The product of two numbers is 192 and their sum is 28. What is the smaller of the two numbers?

Introduction / Context:
When sum and product of two numbers are known, the numbers are roots of the quadratic t^2 − (sum)t + (product) = 0. Solving the quadratic gives both values; choose the smaller as required.

Given Data / Assumptions:

• x + y = 28
• xy = 192

Concept / Approach:
Construct the quadratic: t^2 − 28t + 192 = 0 and solve via factoring or the quadratic formula.

Step-by-Step Solution:
t^2 − 28t + 192 = 0Discriminant = 28^2 − 4*192 = 784 − 768 = 16Roots = (28 ± 4)/2 → 16 and 12Smaller number = 12

Verification / Alternative check:
12 + 16 = 28 and 12 * 16 = 192, both conditions satisfied.

Why Other Options Are Wrong:
14, 18, and 10 do not form a pair with another number to meet both the sum and product simultaneously; 16 is the larger root.

Common Pitfalls:
Arithmetic slips with the discriminant or mixing up which root is smaller.

Final Answer:
12