An algebra problem by Lakshya Kumar

Let $F(x)=3x^2 - 7x+C$ .

where C is variable coefficient and $x>7/6$ .

Then the value of C such That F(x) touches $G(x)=1/(6 )(√(12x+49-12c) +7)$ . then find value of [C] .

$\bullet$ where $[ . ]$ . represents greatest integer function