The sum of a natural number and its square is 156. Find the number.

Let the required number be x

According to given condition,

x + x^{2} = 156

x^{2} + x – 156 = 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 = 1 c = – 156

= 1. – 156 = – 156

And either of their sum or difference = b

= 1

Thus the two terms are 13 and – 12

Sum = 13 – 12 = 1

Product = 13. – 12 = – 156

x^{2} + x – 156 = 0

x^{2} + 13x – 12x – 156 = 0

x(x + 13) – 12 (x + 13) = 0

(x – 12) (x + 13) = 0

x = 12 or x = – 13

x cannot be negative

Hence the required natural number is 12

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