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: 

`x^2-y^2=45`                           ......................(1) 

`y^2=4x`                                  .....................(2)    

From (1) and (2), we get:  

`x^2-4x=45` 

⇒`x^2-4x-45=0` 

⇒`x^2-(9-5)x-45=0` 

⇒`x^2-9x+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 (2), we get:  

`y^2=4xx9` 

⇒`y^2=36` 

⇒`y=6`