F(x+2)=x2+1

Solution for F(x+2)=x2+1 equation:

(F+2)=F2+1

We move all terms to the left:

(F+2)-(F2+1)=0

We add all the numbers together, and all the variables

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

We get rid of parentheses

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

We add all the numbers together, and all the variables

-1F^2+F+1=0

a = -1; b = 1; c = +1;
Δ = b2-4ac
Δ = 12-4·(-1)·1
Δ = 5
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{-(1)-\sqrt{5}}{2*-1}=\frac{-1-\sqrt{5}}{-2}
$
$F_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(1)+\sqrt{5}}{2*-1}=\frac{-1+\sqrt{5}}{-2}
$