1/3x+2/3x-16=1/6x-6

Solution for 1/3x+2/3x-16=1/6x-6 equation:

1/3x+2/3x-16=1/6x-6

We move all terms to the left:

1/3x+2/3x-16-(1/6x-6)=0


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R



Domain of the equation: 6x-6)!=0

x∈R


We get rid of parentheses

1/3x+2/3x-1/6x+6-16=0

We calculate fractions


(12x+1)/18x^2+(-3x)/18x^2+6-16=0

We add all the numbers together, and all the variables

(12x+1)/18x^2+(-3x)/18x^2-10=0

We multiply all the terms by the denominator


(12x+1)+(-3x)-10*18x^2=0

Wy multiply elements

-180x^2+(12x+1)+(-3x)=0

We get rid of parentheses

-180x^2+12x-3x+1=0

We add all the numbers together, and all the variables

-180x^2+9x+1=0

a = -180; b = 9; c = +1;
Δ = b2-4ac
Δ = 92-4·(-180)·1
Δ = 801
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:

$\sqrt{\Delta}=\sqrt{801}=\sqrt{9*89}=\sqrt{9}*\sqrt{89}=3\sqrt{89}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(9)-3\sqrt{89}}{2*-180}=\frac{-9-3\sqrt{89}}{-360}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(9)+3\sqrt{89}}{2*-180}=\frac{-9+3\sqrt{89}}{-360}
$