x+5/6x=3x-14

Solution for x+5/6x=3x-14 equation:

x+5/6x=3x-14

We move all terms to the left:

x+5/6x-(3x-14)=0


Domain of the equation: 6x!=0

x!=0/6

x!=0

x∈R


We get rid of parentheses

x+5/6x-3x+14=0

We multiply all the terms by the denominator


x*6x-3x*6x+14*6x+5=0

Wy multiply elements

6x^2-18x^2+84x+5=0

We add all the numbers together, and all the variables

-12x^2+84x+5=0

a = -12; b = 84; c = +5;
Δ = b2-4ac
Δ = 842-4·(-12)·5
Δ = 7296
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:

$\sqrt{\Delta}=\sqrt{7296}=\sqrt{64*114}=\sqrt{64}*\sqrt{114}=8\sqrt{114}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(84)-8\sqrt{114}}{2*-12}=\frac{-84-8\sqrt{114}}{-24}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(84)+8\sqrt{114}}{2*-12}=\frac{-84+8\sqrt{114}}{-24}
$