Question

22. The difference of the squares of two natural numbers is 45. The square
of the smaller number is four times the larger number. Find the
numbers.
[CBSE 2007]

Answer 1

Let the smaller natural number be x and larger natural number be y

Hence x2 = 4y → (1)

Given y2 – x2 = 45

⇒ y2 – 4y = 45

⇒ y2 – 4y – 45 = 0

⇒ y2 – 9y + 5y – 45 = 0

⇒ y(y – 9) + 5(y – 9) = 0

⇒ (y – 9)(y + 5) = 0

⇒ (y – 9)= 0 or (y + 5) = 0

∴ y = 9 or y = -5

But y is natural number, hence y ≠ - 5

Therefore, y = 9

Equation (1) becomes,

x2 = 4(9) = 36

∴ x = 6

Thus the two natural numbers are 6 and 9.