1/3x+2/3x=x+1

Solution for 1/3x+2/3x=x+1 equation:

1/3x+2/3x=x+1

We move all terms to the left:

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


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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

We add all the numbers together, and all the variables

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

Wy multiply elements

-3x^2-3x+3=0

a = -3; b = -3; c = +3;
Δ = b2-4ac
Δ = -32-4·(-3)·3
Δ = 45
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:

$\sqrt{\Delta}=\sqrt{45}=\sqrt{9*5}=\sqrt{9}*\sqrt{5}=3\sqrt{5}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-3)-3\sqrt{5}}{2*-3}=\frac{3-3\sqrt{5}}{-6}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-3)+3\sqrt{5}}{2*-3}=\frac{3+3\sqrt{5}}{-6}
$