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

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

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

We move all terms to the left:

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


Domain of the equation: 6F+3)!=0

F∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

6F^2-12F^2-18F+1=0

We add all the numbers together, and all the variables

-6F^2-18F+1=0

a = -6; b = -18; c = +1;
Δ = b2-4ac
Δ = -182-4·(-6)·1
Δ = 348
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:

$\sqrt{\Delta}=\sqrt{348}=\sqrt{4*87}=\sqrt{4}*\sqrt{87}=2\sqrt{87}$
$F_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-18)-2\sqrt{87}}{2*-6}=\frac{18-2\sqrt{87}}{-12}
$
$F_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-18)+2\sqrt{87}}{2*-6}=\frac{18+2\sqrt{87}}{-12}
$