The sum of a natural number and its positive square root is 132. Find the number.

Let the required number be x

According to given condition,

putting we get

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 = 1 c = – 132

= 1. – 132 = – 132

And either of their sum or difference = b

= 1

Thus the two terms are 12 and – 11

Difference = 12 – 11 = 1

Product = 12. – 11 = – 132

y(y + 12) – 11(y + 12) = 0

(y + 12) (y – 11) = 0

(y + 12) = 0 or (y – 11) = 0

y = – 12 or y = 11 but y cannot be negative

Thus y = 11

Now

x = y squaring both sides

x = (11)² = 121

Hence the required number is 121