3+2/(x-1)=-x+10

Solution for 3+2/(x-1)=-x+10 equation:

3+2/(x-1)=-x+10

We move all terms to the left:

3+2/(x-1)-(-x+10)=0


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

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

x!=1

x∈R


We add all the numbers together, and all the variables

2/(x-1)-(-1x+10)+3=0

We get rid of parentheses

2/(x-1)+1x-10+3=0

We multiply all the terms by the denominator


1x*(x-1)-10*(x-1)+3*(x-1)+2=0

We multiply parentheses

x^2-1x-10x+3x+10-3+2=0

We add all the numbers together, and all the variables

x^2-8x+9=0

a = 1; b = -8; c = +9;
Δ = b2-4ac
Δ = -82-4·1·9
Δ = 28
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:

$\sqrt{\Delta}=\sqrt{28}=\sqrt{4*7}=\sqrt{4}*\sqrt{7}=2\sqrt{7}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-8)-2\sqrt{7}}{2*1}=\frac{8-2\sqrt{7}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-8)+2\sqrt{7}}{2*1}=\frac{8+2\sqrt{7}}{2}
$