F(5)=4x^2+3

Solution for F(5)=4x^2+3 equation:

(5)=4F^2+3

We move all terms to the left:

(5)-(4F^2+3)=0

We get rid of parentheses

-4F^2-3+5=0

We add all the numbers together, and all the variables

-4F^2+2=0

a = -4; b = 0; c = +2;
Δ = b2-4ac
Δ = 02-4·(-4)·2
Δ = 32
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:

$\sqrt{\Delta}=\sqrt{32}=\sqrt{16*2}=\sqrt{16}*\sqrt{2}=4\sqrt{2}$
$F_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-4\sqrt{2}}{2*-4}=\frac{0-4\sqrt{2}}{-8}
=-\frac{4\sqrt{2}}{-8}
=-\frac{\sqrt{2}}{-2}
$
$F_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+4\sqrt{2}}{2*-4}=\frac{0+4\sqrt{2}}{-8}
=\frac{4\sqrt{2}}{-8}
=\frac{\sqrt{2}}{-2}
$