Two numbers are such that their product is equal to 21 and the sum of their…

Concept

For any two real numbers x and y, the identity (x − y)2 = x2 + y2 − 2xy connects their difference with the sum of their squares and their product.

Because a difference is taken as a magnitude here, the required value is |x − y| = √(x2 + y2 − 2xy).

Application

1. Let the two numbers be x and y. The given data are xy = 21 and x2 + y2 = 58.

2. Substitute these values: (x − y)2 = 58 − 2 × 21.

3. Simplify: (x − y)2 = 58 − 42 = 16.

4. Take the non-negative square root: |x − y| = √16 = 4.

Cross-check

1. The integer pair 3 and 7 has product 21 and square sum 32 + 72 = 9 + 49 = 58; its difference is 7 − 3 = 4.

Therefore, the difference between the two numbers is 4.