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^2-4x+4=0

a = -2; b = -4; c = +4;
Δ = b2-4ac
Δ = -42-4·(-2)·4
Δ = 48
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:

$\sqrt{\Delta}=\sqrt{48}=\sqrt{16*3}=\sqrt{16}*\sqrt{3}=4\sqrt{3}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-4)-4\sqrt{3}}{2*-2}=\frac{4-4\sqrt{3}}{-4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-4)+4\sqrt{3}}{2*-2}=\frac{4+4\sqrt{3}}{-4}
$