3(x-2)-2(x-3)=5x(x-2)-(x-1)

Solution for 3(x-2)-2(x-3)=5x(x-2)-(x-1) equation:

3(x-2)-2(x-3)=5x(x-2)-(x-1)

We move all terms to the left:

3(x-2)-2(x-3)-(5x(x-2)-(x-1))=0

We multiply parentheses

3x-2x-(5x(x-2)-(x-1))-6+6=0


We calculate terms in parentheses: -(5x(x-2)-(x-1)), so:

5x(x-2)-(x-1)

We multiply parentheses

5x^2-10x-(x-1)

We get rid of parentheses

5x^2-10x-x+1

We add all the numbers together, and all the variables

5x^2-11x+1

Back to the equation:

-(5x^2-11x+1)


We add all the numbers together, and all the variables

x-(5x^2-11x+1)=0

We get rid of parentheses

-5x^2+x+11x-1=0

We add all the numbers together, and all the variables

-5x^2+12x-1=0

a = -5; b = 12; c = -1;
Δ = b2-4ac
Δ = 122-4·(-5)·(-1)
Δ = 124
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:

$\sqrt{\Delta}=\sqrt{124}=\sqrt{4*31}=\sqrt{4}*\sqrt{31}=2\sqrt{31}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(12)-2\sqrt{31}}{2*-5}=\frac{-12-2\sqrt{31}}{-10}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(12)+2\sqrt{31}}{2*-5}=\frac{-12+2\sqrt{31}}{-10}
$