1-1/2x=6+x

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

1-1/2x=6+x

We move all terms to the left:

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

We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

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

We add all the numbers together, and all the variables

-2x^2-10x-1=0

a = -2; b = -10; c = -1;
Δ = b2-4ac
Δ = -102-4·(-2)·(-1)
Δ = 92
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:

$\sqrt{\Delta}=\sqrt{92}=\sqrt{4*23}=\sqrt{4}*\sqrt{23}=2\sqrt{23}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-10)-2\sqrt{23}}{2*-2}=\frac{10-2\sqrt{23}}{-4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-10)+2\sqrt{23}}{2*-2}=\frac{10+2\sqrt{23}}{-4}
$