1/6x=3x-68

Solution for 1/6x=3x-68 equation:

1/6x=3x-68

We move all terms to the left:

1/6x-(3x-68)=0


Domain of the equation: 6x!=0

x!=0/6

x!=0

x∈R


We get rid of parentheses

1/6x-3x+68=0

We multiply all the terms by the denominator


-3x*6x+68*6x+1=0

Wy multiply elements

-18x^2+408x+1=0

a = -18; b = 408; c = +1;
Δ = b2-4ac
Δ = 4082-4·(-18)·1
Δ = 166536
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:

$\sqrt{\Delta}=\sqrt{166536}=\sqrt{324*514}=\sqrt{324}*\sqrt{514}=18\sqrt{514}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(408)-18\sqrt{514}}{2*-18}=\frac{-408-18\sqrt{514}}{-36}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(408)+18\sqrt{514}}{2*-18}=\frac{-408+18\sqrt{514}}{-36}
$