1/2w-6=w+7

Solution for 1/2w-6=w+7 equation:

1/2w-6=w+7

We move all terms to the left:

1/2w-6-(w+7)=0


Domain of the equation: 2w!=0

w!=0/2

w!=0

w∈R


We get rid of parentheses

1/2w-w-7-6=0

We multiply all the terms by the denominator


-w*2w-7*2w-6*2w+1=0

Wy multiply elements

-2w^2-14w-12w+1=0

We add all the numbers together, and all the variables

-2w^2-26w+1=0

a = -2; b = -26; c = +1;
Δ = b2-4ac
Δ = -262-4·(-2)·1
Δ = 684
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$w_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$w_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{684}=\sqrt{36*19}=\sqrt{36}*\sqrt{19}=6\sqrt{19}$
$w_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-26)-6\sqrt{19}}{2*-2}=\frac{26-6\sqrt{19}}{-4}
$
$w_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-26)+6\sqrt{19}}{2*-2}=\frac{26+6\sqrt{19}}{-4}
$