G(x)=4x-19/x-5

Solution for G(x)=4x-19/x-5 equation:

(G)=4G-19/G-5

We move all terms to the left:

(G)-(4G-19/G-5)=0


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

G∈R


We get rid of parentheses

G-4G+19/G+5=0

We multiply all the terms by the denominator


G*G-4G*G+5*G+19=0

We add all the numbers together, and all the variables

5G+G*G-4G*G+19=0

Wy multiply elements

G^2-4G^2+5G+19=0

We add all the numbers together, and all the variables

-3G^2+5G+19=0

a = -3; b = 5; c = +19;
Δ = b2-4ac
Δ = 52-4·(-3)·19
Δ = 253
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$G_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$G_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$G_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(5)-\sqrt{253}}{2*-3}=\frac{-5-\sqrt{253}}{-6}
$
$G_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(5)+\sqrt{253}}{2*-3}=\frac{-5+\sqrt{253}}{-6}
$