F(x)=1/3x-2

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

(F)=1/3F-2

We move all terms to the left:

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


Domain of the equation: 3F-2)!=0

F∈R


We get rid of parentheses

F-1/3F+2=0

We multiply all the terms by the denominator


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

Wy multiply elements

3F^2+6F-1=0

a = 3; b = 6; c = -1;
Δ = b2-4ac
Δ = 62-4·3·(-1)
Δ = 48
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:

$\sqrt{\Delta}=\sqrt{48}=\sqrt{16*3}=\sqrt{16}*\sqrt{3}=4\sqrt{3}$
$F_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(6)-4\sqrt{3}}{2*3}=\frac{-6-4\sqrt{3}}{6}
$
$F_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(6)+4\sqrt{3}}{2*3}=\frac{-6+4\sqrt{3}}{6}
$