1/2x-3=2x+1

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

1/2x-3=2x+1

We move all terms to the left:

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


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

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

We add all the numbers together, and all the variables

-4x^2-8x+1=0

a = -4; b = -8; c = +1;
Δ = b2-4ac
Δ = -82-4·(-4)·1
Δ = 80
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:

$\sqrt{\Delta}=\sqrt{80}=\sqrt{16*5}=\sqrt{16}*\sqrt{5}=4\sqrt{5}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-8)-4\sqrt{5}}{2*-4}=\frac{8-4\sqrt{5}}{-8}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-8)+4\sqrt{5}}{2*-4}=\frac{8+4\sqrt{5}}{-8}
$