What is the difference between the smallest perfect square larger than 1 million and the largest perfect square smaller than 1 million?

Answer 1

1 million = 1,000,000 = 1000²

Which means we want: (1000 + 1)² - (1000 - 1)² , that is 1001² - 999²

However, before we work that out, lets look at what we have.

let a = 1000, b = 1, then:

(1000 + 1)² - (1000 - 1)² = (a + b)² - (a - b)²

Now (a + b)² = a² +2ab + b²

and (a - b)² = a² - 2ab - b²

→ (a + b)² - (a - b)² = a² +2ab + b² - (a² - 2ab - b²) = 4ab

As a = 1000, b = 1, this means (1000 + 1)² - (1000 -1)² = 4 × 1000 × 1 = 4000.