x-3=1/2x+2

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

x-3=1/2x+2

We move all terms to the left:

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


Domain of the equation: 2x+2)!=0

x∈R


We get rid of parentheses

x-1/2x-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}
$