1/2x+6=-6x+19

Solution for 1/2x+6=-6x+19 equation:

1/2x+6=-6x+19

We move all terms to the left:

1/2x+6-(-6x+19)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

1/2x+6x-19+6=0

We multiply all the terms by the denominator


6x*2x-19*2x+6*2x+1=0

Wy multiply elements

12x^2-38x+12x+1=0

We add all the numbers together, and all the variables

12x^2-26x+1=0

a = 12; b = -26; c = +1;
Δ = b2-4ac
Δ = -262-4·12·1
Δ = 628
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:

$\sqrt{\Delta}=\sqrt{628}=\sqrt{4*157}=\sqrt{4}*\sqrt{157}=2\sqrt{157}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-26)-2\sqrt{157}}{2*12}=\frac{26-2\sqrt{157}}{24}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-26)+2\sqrt{157}}{2*12}=\frac{26+2\sqrt{157}}{24}
$