Find a natural number whose square diminished . . .

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

Doubt by Muskan 

Solution : 

Let the natural number be x.

ATQ
x²  - 84 = 3 ( 8 + x) 
x² - 84 = 24 + 3x 
x² - 84 - 24 - 3x = 0
x² - 108 - 3x = 0
x² - 3x - 108 = 0 
x² - (12 - 9)x - 108 = 0
x² - 12x + 9x - 108 = 0
x(x - 12) + 9(x - 12) = 0
(x + 9)(x - 12) = 0

x + 9 = 0 OR x - 12 = 0
x = - 9 OR x = 12
x ≠ -9 [∵ x is a natural number] 
∴ x = 12

Hence, required natural number is 12.