2a-8/a=30/a+4

Solution for 2a-8/a=30/a+4 equation:

2a-8/a=30/a+4

We move all terms to the left:

2a-8/a-(30/a+4)=0


Domain of the equation: a!=0

a∈R



Domain of the equation: a+4)!=0

a∈R


We get rid of parentheses

2a-8/a-30/a-4=0

We multiply all the terms by the denominator


2a*a-4*a-8-30=0

We add all the numbers together, and all the variables

-4a+2a*a-38=0

Wy multiply elements

2a^2-4a-38=0

a = 2; b = -4; c = -38;
Δ = b2-4ac
Δ = -42-4·2·(-38)
Δ = 320
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:

$\sqrt{\Delta}=\sqrt{320}=\sqrt{64*5}=\sqrt{64}*\sqrt{5}=8\sqrt{5}$
$a_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-4)-8\sqrt{5}}{2*2}=\frac{4-8\sqrt{5}}{4}
$
$a_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-4)+8\sqrt{5}}{2*2}=\frac{4+8\sqrt{5}}{4}
$