F(3x)=x^2-6

Solution for F(3x)=x^2-6 equation:

(3F)=F^2-6

We move all terms to the left:

(3F)-(F^2-6)=0

We get rid of parentheses

-F^2+3F+6=0

We add all the numbers together, and all the variables

-1F^2+3F+6=0

a = -1; b = 3; c = +6;
Δ = b2-4ac
Δ = 32-4·(-1)·6
Δ = 33
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$F_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$F_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$F_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(3)-\sqrt{33}}{2*-1}=\frac{-3-\sqrt{33}}{-2}
$
$F_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(3)+\sqrt{33}}{2*-1}=\frac{-3+\sqrt{33}}{-2}
$