19+1/2x=5x-1

Solution for 19+1/2x=5x-1 equation:

19+1/2x=5x-1

We move all terms to the left:

19+1/2x-(5x-1)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

1/2x-5x+1+19=0

We multiply all the terms by the denominator


-5x*2x+1*2x+19*2x+1=0

Wy multiply elements

-10x^2+2x+38x+1=0

We add all the numbers together, and all the variables

-10x^2+40x+1=0

a = -10; b = 40; c = +1;
Δ = b2-4ac
Δ = 402-4·(-10)·1
Δ = 1640
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:

$\sqrt{\Delta}=\sqrt{1640}=\sqrt{4*410}=\sqrt{4}*\sqrt{410}=2\sqrt{410}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(40)-2\sqrt{410}}{2*-10}=\frac{-40-2\sqrt{410}}{-20}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(40)+2\sqrt{410}}{2*-10}=\frac{-40+2\sqrt{410}}{-20}
$