1/2x=3x-6

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

1/2x=3x-6

We move all terms to the left:

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


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

1/2x-3x+6=0

We multiply all the terms by the denominator


-3x*2x+6*2x+1=0

Wy multiply elements

-6x^2+12x+1=0

a = -6; b = 12; c = +1;
Δ = b2-4ac
Δ = 122-4·(-6)·1
Δ = 168
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:

$\sqrt{\Delta}=\sqrt{168}=\sqrt{4*42}=\sqrt{4}*\sqrt{42}=2\sqrt{42}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(12)-2\sqrt{42}}{2*-6}=\frac{-12-2\sqrt{42}}{-12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(12)+2\sqrt{42}}{2*-6}=\frac{-12+2\sqrt{42}}{-12}
$