3+1/3x=-2x+31

Solution for 3+1/3x=-2x+31 equation:

3+1/3x=-2x+31

We move all terms to the left:

3+1/3x-(-2x+31)=0


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R


We get rid of parentheses

1/3x+2x-31+3=0

We multiply all the terms by the denominator


2x*3x-31*3x+3*3x+1=0

Wy multiply elements

6x^2-93x+9x+1=0

We add all the numbers together, and all the variables

6x^2-84x+1=0

a = 6; b = -84; c = +1;
Δ = b2-4ac
Δ = -842-4·6·1
Δ = 7032
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:

$\sqrt{\Delta}=\sqrt{7032}=\sqrt{4*1758}=\sqrt{4}*\sqrt{1758}=2\sqrt{1758}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-84)-2\sqrt{1758}}{2*6}=\frac{84-2\sqrt{1758}}{12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-84)+2\sqrt{1758}}{2*6}=\frac{84+2\sqrt{1758}}{12}
$