F(x)=4x-2/6-x^2

Solution for F(x)=4x-2/6-x^2 equation:

(F)=4F-2/6-F^2

We move all terms to the left:

(F)-(4F-2/6-F^2)=0


Domain of the equation: 6-F^2)!=0

We move all terms containing F to the left, all other terms to the right

-F^2)!=-6

F!=-6/1

F!=-6

F∈R


We get rid of parentheses

F^2-4F+F+2/6=0

We multiply all the terms by the denominator


F^2*6-4F*6+F*6+2=0

Wy multiply elements

6F^2-24F+6F+2=0

We add all the numbers together, and all the variables

6F^2-18F+2=0

a = 6; b = -18; c = +2;
Δ = b2-4ac
Δ = -182-4·6·2
Δ = 276
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:

$\sqrt{\Delta}=\sqrt{276}=\sqrt{4*69}=\sqrt{4}*\sqrt{69}=2\sqrt{69}$
$F_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-18)-2\sqrt{69}}{2*6}=\frac{18-2\sqrt{69}}{12}
$
$F_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-18)+2\sqrt{69}}{2*6}=\frac{18+2\sqrt{69}}{12}
$