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

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

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

We move all terms to the left:

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


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

-8x^2+4x+6x-1=0

We add all the numbers together, and all the variables

-8x^2+10x-1=0

a = -8; b = 10; c = -1;
Δ = b2-4ac
Δ = 102-4·(-8)·(-1)
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{-(10)-2\sqrt{17}}{2*-8}=\frac{-10-2\sqrt{17}}{-16}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(10)+2\sqrt{17}}{2*-8}=\frac{-10+2\sqrt{17}}{-16}
$