1/2x+8=6x-12-13/2x

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Solution for 1/2x+8=6x-12-13/2x equation:



1/2x+8=6x-12-13/2x
We move all terms to the left:
1/2x+8-(6x-12-13/2x)=0
Domain of the equation: 2x!=0
x!=0/2
x!=0
x∈R
Domain of the equation: 2x)!=0
x!=0/1
x!=0
x∈R
We add all the numbers together, and all the variables
1/2x-(6x-13/2x-12)+8=0
We get rid of parentheses
1/2x-6x+13/2x+12+8=0
We multiply all the terms by the denominator
-6x*2x+12*2x+8*2x+1+13=0
We add all the numbers together, and all the variables
-6x*2x+12*2x+8*2x+14=0
Wy multiply elements
-12x^2+24x+16x+14=0
We add all the numbers together, and all the variables
-12x^2+40x+14=0
a = -12; b = 40; c = +14;
Δ = b2-4ac
Δ = 402-4·(-12)·14
Δ = 2272
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:
$\sqrt{\Delta}=\sqrt{2272}=\sqrt{16*142}=\sqrt{16}*\sqrt{142}=4\sqrt{142}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(40)-4\sqrt{142}}{2*-12}=\frac{-40-4\sqrt{142}}{-24} $
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(40)+4\sqrt{142}}{2*-12}=\frac{-40+4\sqrt{142}}{-24} $

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