1/3x+3=7-2x

Solution for 1/3x+3=7-2x equation:

1/3x+3=7-2x

We move all terms to the left:

1/3x+3-(7-2x)=0


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R


We add all the numbers together, and all the variables

1/3x-(-2x+7)+3=0

We get rid of parentheses

1/3x+2x-7+3=0

We multiply all the terms by the denominator


2x*3x-7*3x+3*3x+1=0

Wy multiply elements

6x^2-21x+9x+1=0

We add all the numbers together, and all the variables

6x^2-12x+1=0

a = 6; b = -12; c = +1;
Δ = b2-4ac
Δ = -122-4·6·1
Δ = 120
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:

$\sqrt{\Delta}=\sqrt{120}=\sqrt{4*30}=\sqrt{4}*\sqrt{30}=2\sqrt{30}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-12)-2\sqrt{30}}{2*6}=\frac{12-2\sqrt{30}}{12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-12)+2\sqrt{30}}{2*6}=\frac{12+2\sqrt{30}}{12}
$