Let n be a required natural number.
Square of a natural number diminished by 84 = n2 – 84
and thrice of 8 more than the natural number = 3 (n + 8)
Now, by given condition,
n2 – 84 = 3 (n + 8)
⇒ n2 – 84 = 3n + 24
⇒ n2 – 3n – 108 = 0
⇒ n2 – 12n + 9n – 108 = 0 [by splitting the middle term]
⇒ n (n – 12) + 9 (n – 12) = 0
⇒ (n – 12) (n + 9) = 0
⇒ n = 12 [n ≠ – 9 because n is a natural number
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