Question

The difference of the square of two natural numbers is 45. The square of the smaller number is 4 times the larger number. Determine the numbers.

Answer 1

Let these two numbers be X and Y, Y being the bigger number. Then as per the question, 

Y² - X² =45 ..... (i) 

X²  = 4Y ..... (ii) 

From (i), we get X2=Y2- 45, Putting this in (ii), we get 

Y² - 45 - 4Y = O 

⇒ Y² - 4Y- 45 = 0 

⇒ Y² - 9Y + 5Y - 45 = 0 

⇒ Y (Y - 9) + 5( Y - 9) = 0 

⇒ (Y-9) (Y+S) = 0

⇒  Y can't be a negative number and hence Y=9 

⇒ Putting Y = 9 in (ii), we get X² = 4 x 9=36 

⇒ X = 6

Hence numbers are 6 and 9.