1/2x+19=10x-19

Solution for 1/2x+19=10x-19 equation:

1/2x+19=10x-19

We move all terms to the left:

1/2x+19-(10x-19)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

1/2x-10x+19+19=0

We multiply all the terms by the denominator


-10x*2x+19*2x+19*2x+1=0

Wy multiply elements

-20x^2+38x+38x+1=0

We add all the numbers together, and all the variables

-20x^2+76x+1=0

a = -20; b = 76; c = +1;
Δ = b2-4ac
Δ = 762-4·(-20)·1
Δ = 5856
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:

$\sqrt{\Delta}=\sqrt{5856}=\sqrt{16*366}=\sqrt{16}*\sqrt{366}=4\sqrt{366}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(76)-4\sqrt{366}}{2*-20}=\frac{-76-4\sqrt{366}}{-40}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(76)+4\sqrt{366}}{2*-20}=\frac{-76+4\sqrt{366}}{-40}
$