x-6=1/2x-2

Solution for x-6=1/2x-2 equation:

x-6=1/2x-2

We move all terms to the left:

x-6-(1/2x-2)=0


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

x∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


x*2x+2*2x-6*2x-1=0

Wy multiply elements

2x^2+4x-12x-1=0

We add all the numbers together, and all the variables

2x^2-8x-1=0

a = 2; b = -8; c = -1;
Δ = b2-4ac
Δ = -82-4·2·(-1)
Δ = 72
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:

$\sqrt{\Delta}=\sqrt{72}=\sqrt{36*2}=\sqrt{36}*\sqrt{2}=6\sqrt{2}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-8)-6\sqrt{2}}{2*2}=\frac{8-6\sqrt{2}}{4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-8)+6\sqrt{2}}{2*2}=\frac{8+6\sqrt{2}}{4}
$