Q. 225.0( 5 Votes )

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 :

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

According to the question


x2 – y2 = 45 – – – – – (1)


y2 = 4x – – – – – – (2)


From (1) and (2) we get


x2 – 4x = 45


x2 – 4x – 45 = 0


Using the splitting middle term – the middle term of the general equation is divided in two such values that:


Product = a.c


For the given equation a = 1 b = – 4 c = – 45


= 1. – 45 = – 45


And either of their sum or difference = b


= – 4


Thus the two terms are – 9 and 5


Sum = – 9 + 5 = – 4


Product = – 9.5 = – 45


x2 – 9x + 5x – 45 = 0


x(x – 9) + 5(x – 9) = 0


(x + 5) (x – 9) = 0


(x + 5) = 0 or (x – 9) = 0


x = – 5 or x = 9


x = 9


putting the value of x in equation (2), we get


y2 = 4.9 = 36


taking square root


y = 6


Hence the two numbers are 9 and 6


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