#### Find a natural number whose square diminished by 84 is equal to thrice of 8 more than the given number

12

Solution

Let the natural number = x

It is given that when the square of x is diminished by 84 then it is equal to thrice of 8 more than x
$x^{2}-84=3(x+8)\\ x^{2}-84=3x+24\\ x^{2}-3x-84-24=0\\ x^{2}-3x-108=0\\ x^{2}-12x+9x-108=0\\ x(x-12)+9(x-12)=0\\ (x-12)(x+9)=0$

x = 12, – 9

We can’t take – 9 because it is given that x is a natural number.

Hence x = 12 is the required natural number.