F(3)=3x^2-6x+1

Solution for F(3)=3x^2-6x+1 equation:

(3)=3F^2-6F+1

We move all terms to the left:

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

We get rid of parentheses

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

We add all the numbers together, and all the variables

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

a = -3; b = 6; c = +2;
Δ = b2-4ac
Δ = 62-4·(-3)·2
Δ = 60
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$F_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$F_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{60}=\sqrt{4*15}=\sqrt{4}*\sqrt{15}=2\sqrt{15}$
$F_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(6)-2\sqrt{15}}{2*-3}=\frac{-6-2\sqrt{15}}{-6}
$
$F_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(6)+2\sqrt{15}}{2*-3}=\frac{-6+2\sqrt{15}}{-6}
$