2/3x-5+2(x-3)=12x+7

Solution for 2/3x-5+2(x-3)=12x+7 equation:

2/3x-5+2(x-3)=12x+7

We move all terms to the left:

2/3x-5+2(x-3)-(12x+7)=0


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R


We multiply parentheses

2/3x+2x-(12x+7)-6-5=0

We get rid of parentheses

2/3x+2x-12x-7-6-5=0

We multiply all the terms by the denominator


2x*3x-12x*3x-7*3x-6*3x-5*3x+2=0

Wy multiply elements

6x^2-36x^2-21x-18x-15x+2=0

We add all the numbers together, and all the variables

-30x^2-54x+2=0

a = -30; b = -54; c = +2;
Δ = b2-4ac
Δ = -542-4·(-30)·2
Δ = 3156
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:

$\sqrt{\Delta}=\sqrt{3156}=\sqrt{4*789}=\sqrt{4}*\sqrt{789}=2\sqrt{789}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-54)-2\sqrt{789}}{2*-30}=\frac{54-2\sqrt{789}}{-60}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-54)+2\sqrt{789}}{2*-30}=\frac{54+2\sqrt{789}}{-60}
$