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

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

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

We move all terms to the left:

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

We use the square of the difference formula

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

We get rid of parentheses

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

We add all the numbers together, and all the variables

-1G^2-4G+3=0

a = -1; b = -4; c = +3;
Δ = b2-4ac
Δ = -42-4·(-1)·3
Δ = 28
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:

$\sqrt{\Delta}=\sqrt{28}=\sqrt{4*7}=\sqrt{4}*\sqrt{7}=2\sqrt{7}$
$G_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-4)-2\sqrt{7}}{2*-1}=\frac{4-2\sqrt{7}}{-2}
$
$G_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-4)+2\sqrt{7}}{2*-1}=\frac{4+2\sqrt{7}}{-2}
$