-1/2x+3/2x=4x+1

Solution for -1/2x+3/2x=4x+1 equation:

-1/2x+3/2x=4x+1

We move all terms to the left:

-1/2x+3/2x-(4x+1)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

-1/2x+3/2x-4x-1=0

We multiply all the terms by the denominator


-4x*2x-1*2x-1+3=0

We add all the numbers together, and all the variables

-4x*2x-1*2x+2=0

Wy multiply elements

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

a = -8; b = -2; c = +2;
Δ = b2-4ac
Δ = -22-4·(-8)·2
Δ = 68
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:

$\sqrt{\Delta}=\sqrt{68}=\sqrt{4*17}=\sqrt{4}*\sqrt{17}=2\sqrt{17}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-2)-2\sqrt{17}}{2*-8}=\frac{2-2\sqrt{17}}{-16}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-2)+2\sqrt{17}}{2*-8}=\frac{2+2\sqrt{17}}{-16}
$