Question

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.

Answer 1

Let the greater number be x and the smaller number be y. 

According to the question:

x² - y² = 45.....(i)

y² = 4x.......(ii)

From (i) and (ii), we get:

⇒ x = 9 (∵ x is a natural number)

Putting the value of x in equation (ii), we get:

y² = 4 x 9

⇒ y² = 36

⇒ y = 6

Hence, the two numbers are 9 and 6.

Answer 2

Let the two numbers be x and y such that y < x.

From the given conditions, we have:

x² - y² = 45 … (1)

y² = 4x … (2)

Using (2) in (1), we have:

x² - 4x - 45 = 0

 (x - 9)(x + 5) = 0

 x = 9 (Rejecting the negative value of x)

From (2), we get, y = 6 or -6

Rejecting the negative value, we have, y = 6.

Thus, the two numbers are 9 and 6.