2a-1/3a=180-80

Solution for 2a-1/3a=180-80 equation:

2a-1/3a=180-80

We move all terms to the left:

2a-1/3a-(180-80)=0


Domain of the equation: 3a!=0

a!=0/3

a!=0

a∈R


We add all the numbers together, and all the variables

2a-1/3a-100=0

We multiply all the terms by the denominator


2a*3a-100*3a-1=0

Wy multiply elements

6a^2-300a-1=0

a = 6; b = -300; c = -1;
Δ = b2-4ac
Δ = -3002-4·6·(-1)
Δ = 90024
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:

$\sqrt{\Delta}=\sqrt{90024}=\sqrt{484*186}=\sqrt{484}*\sqrt{186}=22\sqrt{186}$
$a_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-300)-22\sqrt{186}}{2*6}=\frac{300-22\sqrt{186}}{12}
$
$a_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-300)+22\sqrt{186}}{2*6}=\frac{300+22\sqrt{186}}{12}
$