6x=1/2x-1

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

6x=1/2x-1

We move all terms to the left:

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


Domain of the equation: 2x-1)!=0

x∈R


We get rid of parentheses

6x-1/2x+1=0

We multiply all the terms by the denominator


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

Wy multiply elements

12x^2+2x-1=0

a = 12; b = 2; c = -1;
Δ = b2-4ac
Δ = 22-4·12·(-1)
Δ = 52
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:

$\sqrt{\Delta}=\sqrt{52}=\sqrt{4*13}=\sqrt{4}*\sqrt{13}=2\sqrt{13}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(2)-2\sqrt{13}}{2*12}=\frac{-2-2\sqrt{13}}{24}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(2)+2\sqrt{13}}{2*12}=\frac{-2+2\sqrt{13}}{24}
$