Let the larger and the smaller parts be x and y respectively.

According to the question

x + y = 16 – – – – – (1)

2x² = y² + 164 – – – (2)

From (1) x = 16 – y – – – (3)

From (2) and (3) we get

2(16 – y)² = y² + 164

2(256 – 32y + y²) = y² + 164 using (a + b)² = a² + 2ab + b²

512 – 64y + 2y² = y² + 164

y² – 64y + 348 = 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 = – 64 c = 348

= 1. 348 = 348

And either of their sum or difference = b

= – 64

Thus the two terms are – 58 and – 6

Sum = – 58 – 6 = – 64

Product = – 58. – 6 = 348

y² – 64y + 348 = 0

y² – 58y – 6y + 348 = 0

y(y – 58) – 6(y – 58) = 0

(y – 58) (y – 6) = 0

(y – 58) = 0 or (y – 6) = 0

y = 6 (y < 16)

putting the value of y in (3), we get

x = 16 – 6

= 10

Hence the two natural numbers are 6 and 10.