1/2x-1=3x+1

Solution for 1/2x-1=3x+1 equation:

1/2x-1=3x+1

We move all terms to the left:

1/2x-1-(3x+1)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

1/2x-3x-1-1=0

We multiply all the terms by the denominator


-3x*2x-1*2x-1*2x+1=0

Wy multiply elements

-6x^2-2x-2x+1=0

We add all the numbers together, and all the variables

-6x^2-4x+1=0

a = -6; b = -4; c = +1;
Δ = b2-4ac
Δ = -42-4·(-6)·1
Δ = 40
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:

$\sqrt{\Delta}=\sqrt{40}=\sqrt{4*10}=\sqrt{4}*\sqrt{10}=2\sqrt{10}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-4)-2\sqrt{10}}{2*-6}=\frac{4-2\sqrt{10}}{-12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-4)+2\sqrt{10}}{2*-6}=\frac{4+2\sqrt{10}}{-12}
$