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.

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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