1/4x-1=19x+19

Solution for 1/4x-1=19x+19 equation:

1/4x-1=19x+19

We move all terms to the left:

1/4x-1-(19x+19)=0


Domain of the equation: 4x!=0

x!=0/4

x!=0

x∈R


We get rid of parentheses

1/4x-19x-19-1=0

We multiply all the terms by the denominator


-19x*4x-19*4x-1*4x+1=0

Wy multiply elements

-76x^2-76x-4x+1=0

We add all the numbers together, and all the variables

-76x^2-80x+1=0

a = -76; b = -80; c = +1;
Δ = b2-4ac
Δ = -802-4·(-76)·1
Δ = 6704
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{6704}=\sqrt{16*419}=\sqrt{16}*\sqrt{419}=4\sqrt{419}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-80)-4\sqrt{419}}{2*-76}=\frac{80-4\sqrt{419}}{-152}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-80)+4\sqrt{419}}{2*-76}=\frac{80+4\sqrt{419}}{-152}
$