(2a-3/a-3)-2=12/a+3

Solution for (2a-3/a-3)-2=12/a+3 equation:

(2a-3/a-3)-2=12/a+3

We move all terms to the left:

(2a-3/a-3)-2-(12/a+3)=0


Domain of the equation: a-3)!=0

a∈R



Domain of the equation: a+3)!=0

a∈R


We get rid of parentheses

2a-3/a-12/a-3-3-2=0

We multiply all the terms by the denominator


2a*a-3*a-3*a-2*a-3-12=0

We add all the numbers together, and all the variables

-8a+2a*a-15=0

Wy multiply elements

2a^2-8a-15=0

a = 2; b = -8; c = -15;
Δ = b2-4ac
Δ = -82-4·2·(-15)
Δ = 184
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:

$\sqrt{\Delta}=\sqrt{184}=\sqrt{4*46}=\sqrt{4}*\sqrt{46}=2\sqrt{46}$
$a_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-8)-2\sqrt{46}}{2*2}=\frac{8-2\sqrt{46}}{4}
$
$a_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-8)+2\sqrt{46}}{2*2}=\frac{8+2\sqrt{46}}{4}
$