F(3x)=2(3x)^2-1

Solution for F(3x)=2(3x)^2-1 equation:

(3F)=2(3F)^2-1

We move all terms to the left:

(3F)-(2(3F)^2-1)=0

We get rid of parentheses

-23F^2+3F+1=0

a = -23; b = 3; c = +1;
Δ = b2-4ac
Δ = 32-4·(-23)·1
Δ = 101
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{101}}{2*-23}=\frac{-3-\sqrt{101}}{-46}
$
$F_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(3)+\sqrt{101}}{2*-23}=\frac{-3+\sqrt{101}}{-46}
$