x-4/x-1=12/3-x+1

Solution for x-4/x-1=12/3-x+1 equation:

x-4/x-1=12/3-x+1

We move all terms to the left:

x-4/x-1-(12/3-x+1)=0


Domain of the equation: x!=0

x∈R



Domain of the equation: 3-x+1)!=0

We move all terms containing x to the left, all other terms to the right

-x+1)!=-3

x∈R


We add all the numbers together, and all the variables

x-4/x-(-1x+5)-1=0

We get rid of parentheses

x-4/x+1x-5-1=0

We multiply all the terms by the denominator


x*x+1x*x-5*x-1*x-4=0

We add all the numbers together, and all the variables

-6x+x*x+1x*x-4=0

Wy multiply elements

x^2+x^2-6x-4=0

We add all the numbers together, and all the variables

2x^2-6x-4=0

a = 2; b = -6; c = -4;
Δ = b2-4ac
Δ = -62-4·2·(-4)
Δ = 68
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:

$\sqrt{\Delta}=\sqrt{68}=\sqrt{4*17}=\sqrt{4}*\sqrt{17}=2\sqrt{17}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-6)-2\sqrt{17}}{2*2}=\frac{6-2\sqrt{17}}{4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-6)+2\sqrt{17}}{2*2}=\frac{6+2\sqrt{17}}{4}
$