6x+(1/2)(12x-8)=3x-31

Solution for 6x+(1/2)(12x-8)=3x-31 equation:

6x+(1/2)(12x-8)=3x-31

We move all terms to the left:

6x+(1/2)(12x-8)-(3x-31)=0


Domain of the equation: 2)(12x-8)!=0

x∈R


We add all the numbers together, and all the variables

6x+(+1/2)(12x-8)-(3x-31)=0

We get rid of parentheses

6x+(+1/2)(12x-8)-3x+31=0

We multiply parentheses ..

(+12x^2+1/2*-8)+6x-3x+31=0

We multiply all the terms by the denominator


(+12x^2+1+6x*2*-8)-3x*2*-8)+31*2*-8)=0

We add all the numbers together, and all the variables

(+12x^2+1+6x*2*-8)-3x*2*=0

Wy multiply elements

(+12x^2+1+6x*2*-8)-6x^2=0

We get rid of parentheses

12x^2-6x^2+6x*2*+1-8=0

We add all the numbers together, and all the variables

6x^2+6x*2*-7=0

Wy multiply elements

6x^2+12x^2-7=0

We add all the numbers together, and all the variables

18x^2-7=0

a = 18; b = 0; c = -7;
Δ = b2-4ac
Δ = 02-4·18·(-7)
Δ = 504
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:

$\sqrt{\Delta}=\sqrt{504}=\sqrt{36*14}=\sqrt{36}*\sqrt{14}=6\sqrt{14}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-6\sqrt{14}}{2*18}=\frac{0-6\sqrt{14}}{36}
=-\frac{6\sqrt{14}}{36}
=-\frac{\sqrt{14}}{6}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+6\sqrt{14}}{2*18}=\frac{0+6\sqrt{14}}{36}
=\frac{6\sqrt{14}}{36}
=\frac{\sqrt{14}}{6}
$