2(x+3/2x-1)=3(x^2-1)

Solution for 2(x+3/2x-1)=3(x^2-1) equation:

2(x+3/2x-1)=3(x^2-1)

We move all terms to the left:

2(x+3/2x-1)-(3(x^2-1))=0


Domain of the equation: 2x-1)!=0

x∈R


We multiply parentheses

2x+6x-(3(x^2-1))-2=0


We calculate terms in parentheses: -(3(x^2-1)), so:

3(x^2-1)

We multiply parentheses

3x^2-3

Back to the equation:

-(3x^2-3)


We add all the numbers together, and all the variables

8x-(3x^2-3)-2=0

We get rid of parentheses

-3x^2+8x+3-2=0

We add all the numbers together, and all the variables

-3x^2+8x+1=0

a = -3; b = 8; c = +1;
Δ = b2-4ac
Δ = 82-4·(-3)·1
Δ = 76
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{76}=\sqrt{4*19}=\sqrt{4}*\sqrt{19}=2\sqrt{19}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(8)-2\sqrt{19}}{2*-3}=\frac{-8-2\sqrt{19}}{-6}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(8)+2\sqrt{19}}{2*-3}=\frac{-8+2\sqrt{19}}{-6}
$