3a-3=1/2a-23

Solution for 3a-3=1/2a-23 equation:

3a-3=1/2a-23

We move all terms to the left:

3a-3-(1/2a-23)=0


Domain of the equation: 2a-23)!=0

a∈R


We get rid of parentheses

3a-1/2a+23-3=0

We multiply all the terms by the denominator


3a*2a+23*2a-3*2a-1=0

Wy multiply elements

6a^2+46a-6a-1=0

We add all the numbers together, and all the variables

6a^2+40a-1=0

a = 6; b = 40; c = -1;
Δ = b2-4ac
Δ = 402-4·6·(-1)
Δ = 1624
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:

$\sqrt{\Delta}=\sqrt{1624}=\sqrt{4*406}=\sqrt{4}*\sqrt{406}=2\sqrt{406}$
$a_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(40)-2\sqrt{406}}{2*6}=\frac{-40-2\sqrt{406}}{12}
$
$a_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(40)+2\sqrt{406}}{2*6}=\frac{-40+2\sqrt{406}}{12}
$