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

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

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

We move all terms to the left:

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

We get rid of parentheses

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

We multiply all the terms by the denominator


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

We add all the numbers together, and all the variables

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

Wy multiply elements

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

We add all the numbers together, and all the variables

-2x^2+4=0

a = -2; b = 0; c = +4;
Δ = b2-4ac
Δ = 02-4·(-2)·4
Δ = 32
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:

$\sqrt{\Delta}=\sqrt{32}=\sqrt{16*2}=\sqrt{16}*\sqrt{2}=4\sqrt{2}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-4\sqrt{2}}{2*-2}=\frac{0-4\sqrt{2}}{-4}
=-\frac{4\sqrt{2}}{-4}
=-\frac{\sqrt{2}}{-1}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+4\sqrt{2}}{2*-2}=\frac{0+4\sqrt{2}}{-4}
=\frac{4\sqrt{2}}{-4}
=\frac{\sqrt{2}}{-1}
$