1/2x=6+x

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

1/2x=6+x

We move all terms to the left:

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


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We add all the numbers together, and all the variables

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

We get rid of parentheses

1/2x-x-6=0

We multiply all the terms by the denominator


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

Wy multiply elements

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

a = -2; b = -12; c = +1;
Δ = b2-4ac
Δ = -122-4·(-2)·1
Δ = 152
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:

$\sqrt{\Delta}=\sqrt{152}=\sqrt{4*38}=\sqrt{4}*\sqrt{38}=2\sqrt{38}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-12)-2\sqrt{38}}{2*-2}=\frac{12-2\sqrt{38}}{-4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-12)+2\sqrt{38}}{2*-2}=\frac{12+2\sqrt{38}}{-4}
$