31/3x+12=7x+6

Solution for 31/3x+12=7x+6 equation:

31/3x+12=7x+6

We move all terms to the left:

31/3x+12-(7x+6)=0


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R


We get rid of parentheses

31/3x-7x-6+12=0

We multiply all the terms by the denominator


-7x*3x-6*3x+12*3x+31=0

Wy multiply elements

-21x^2-18x+36x+31=0

We add all the numbers together, and all the variables

-21x^2+18x+31=0

a = -21; b = 18; c = +31;
Δ = b2-4ac
Δ = 182-4·(-21)·31
Δ = 2928
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:

$\sqrt{\Delta}=\sqrt{2928}=\sqrt{16*183}=\sqrt{16}*\sqrt{183}=4\sqrt{183}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(18)-4\sqrt{183}}{2*-21}=\frac{-18-4\sqrt{183}}{-42}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(18)+4\sqrt{183}}{2*-21}=\frac{-18+4\sqrt{183}}{-42}
$