1/2x+2-3/2x=1.5x-6

Solution for 1/2x+2-3/2x=1.5x-6 equation:

1/2x+2-3/2x=1.5x-6

We move all terms to the left:

1/2x+2-3/2x-(1.5x-6)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

1/2x-3/2x-1.5x+6+2=0

We multiply all the terms by the denominator


-(1.5x)*2x+6*2x+2*2x+1-3=0

We add all the numbers together, and all the variables

-(+1.5x)*2x+6*2x+2*2x+1-3=0

We add all the numbers together, and all the variables

-(+1.5x)*2x+6*2x+2*2x-2=0

We multiply parentheses

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

Wy multiply elements

-2x^2+12x+4x-2=0

We add all the numbers together, and all the variables

-2x^2+16x-2=0

a = -2; b = 16; c = -2;
Δ = b2-4ac
Δ = 162-4·(-2)·(-2)
Δ = 240
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:

$\sqrt{\Delta}=\sqrt{240}=\sqrt{16*15}=\sqrt{16}*\sqrt{15}=4\sqrt{15}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(16)-4\sqrt{15}}{2*-2}=\frac{-16-4\sqrt{15}}{-4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(16)+4\sqrt{15}}{2*-2}=\frac{-16+4\sqrt{15}}{-4}
$