G(5n)=5n^2-5n+3

Solution for G(5n)=5n^2-5n+3 equation:

(5G)=5G^2-5G+3

We move all terms to the left:

(5G)-(5G^2-5G+3)=0

We get rid of parentheses

-5G^2+5G+5G-3=0

We add all the numbers together, and all the variables

-5G^2+10G-3=0

a = -5; b = 10; c = -3;
Δ = b2-4ac
Δ = 102-4·(-5)·(-3)
Δ = 40
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$G_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$G_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{40}=\sqrt{4*10}=\sqrt{4}*\sqrt{10}=2\sqrt{10}$
$G_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(10)-2\sqrt{10}}{2*-5}=\frac{-10-2\sqrt{10}}{-10}
$
$G_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(10)+2\sqrt{10}}{2*-5}=\frac{-10+2\sqrt{10}}{-10}
$