Answer :

Let the larger number be x and 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

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

x^{2} – 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

y^{2} = 4.9 = 36

taking square root

y = 6

Hence the two numbers are 9 and 6

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