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}
$