1/2x+2=3x-3

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

1/2x+2=3x-3

We move all terms to the left:

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


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

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

We add all the numbers together, and all the variables

-6x^2+10x+1=0

a = -6; b = 10; c = +1;
Δ = b2-4ac
Δ = 102-4·(-6)·1
Δ = 124
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:

$\sqrt{\Delta}=\sqrt{124}=\sqrt{4*31}=\sqrt{4}*\sqrt{31}=2\sqrt{31}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(10)-2\sqrt{31}}{2*-6}=\frac{-10-2\sqrt{31}}{-12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(10)+2\sqrt{31}}{2*-6}=\frac{-10+2\sqrt{31}}{-12}
$