1/2x=1+3x

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

1/2x=1+3x

We move all terms to the left:

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


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We add all the numbers together, and all the variables

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

We get rid of parentheses

1/2x-3x-1=0

We multiply all the terms by the denominator


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

Wy multiply elements

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

a = -6; b = -2; c = +1;
Δ = b2-4ac
Δ = -22-4·(-6)·1
Δ = 28
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:

$\sqrt{\Delta}=\sqrt{28}=\sqrt{4*7}=\sqrt{4}*\sqrt{7}=2\sqrt{7}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-2)-2\sqrt{7}}{2*-6}=\frac{2-2\sqrt{7}}{-12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-2)+2\sqrt{7}}{2*-6}=\frac{2+2\sqrt{7}}{-12}
$