F(z)=2z+1/z-5

Solution for F(z)=2z+1/z-5 equation:

(F)=2F+1/F-5

We move all terms to the left:

(F)-(2F+1/F-5)=0


Domain of the equation: F-5)!=0

F∈R


We get rid of parentheses

F-2F-1/F+5=0

We multiply all the terms by the denominator


F*F-2F*F+5*F-1=0

We add all the numbers together, and all the variables

5F+F*F-2F*F-1=0

Wy multiply elements

F^2-2F^2+5F-1=0

We add all the numbers together, and all the variables

-1F^2+5F-1=0

a = -1; b = 5; c = -1;
Δ = b2-4ac
Δ = 52-4·(-1)·(-1)
Δ = 21
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{-(5)-\sqrt{21}}{2*-1}=\frac{-5-\sqrt{21}}{-2}
$
$F_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(5)+\sqrt{21}}{2*-1}=\frac{-5+\sqrt{21}}{-2}
$