1/2x-3=x+2

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

1/2x-3=x+2

We move all terms to the left:

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


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

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

We add all the numbers together, and all the variables

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

a = -2; b = -10; c = +1;
Δ = b2-4ac
Δ = -102-4·(-2)·1
Δ = 108
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:

$\sqrt{\Delta}=\sqrt{108}=\sqrt{36*3}=\sqrt{36}*\sqrt{3}=6\sqrt{3}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-10)-6\sqrt{3}}{2*-2}=\frac{10-6\sqrt{3}}{-4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-10)+6\sqrt{3}}{2*-2}=\frac{10+6\sqrt{3}}{-4}
$