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:  

x² – 4x = 45 

⇒ x² – 4x – 45 = 0 

⇒ x² – (9 – 5)x – 45 = 0 

⇒ x² – 9x + 5x – 45 = 0 

⇒ x(x – 9) + 5(x – 9) = 0

⇒ (x – 9)(x + 5) = 0 

⇒ x – 9 = 0 or x + 5 = 0 

⇒ x = 9 or x = –5 

⇒ x = 9   ...(∵ x is a natural number)

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

y² = 4 × 9 

⇒ y² = 36 

⇒ y = 6

Hence, the two numbers are 9 and 6.