1/2a-6=3+a

Solution for 1/2a-6=3+a equation:

1/2a-6=3+a

We move all terms to the left:

1/2a-6-(3+a)=0


Domain of the equation: 2a!=0

a!=0/2

a!=0

a∈R


We add all the numbers together, and all the variables

1/2a-(a+3)-6=0

We get rid of parentheses

1/2a-a-3-6=0

We multiply all the terms by the denominator


-a*2a-3*2a-6*2a+1=0

Wy multiply elements

-2a^2-6a-12a+1=0

We add all the numbers together, and all the variables

-2a^2-18a+1=0

a = -2; b = -18; c = +1;
Δ = b2-4ac
Δ = -182-4·(-2)·1
Δ = 332
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$a_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$a_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{332}=\sqrt{4*83}=\sqrt{4}*\sqrt{83}=2\sqrt{83}$
$a_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-18)-2\sqrt{83}}{2*-2}=\frac{18-2\sqrt{83}}{-4}
$
$a_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-18)+2\sqrt{83}}{2*-2}=\frac{18+2\sqrt{83}}{-4}
$