2/(x-3)+4=2x

Solution for 2/(x-3)+4=2x equation:

2/(x-3)+4=2x

We move all terms to the left:

2/(x-3)+4-(2x)=0


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

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

x!=3

x∈R


We add all the numbers together, and all the variables

-2x+2/(x-3)+4=0

We multiply all the terms by the denominator


-2x*(x-3)+4*(x-3)+2=0

We multiply parentheses

-2x^2+6x+4x-12+2=0

We add all the numbers together, and all the variables

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

a = -2; b = 10; c = -10;
Δ = b2-4ac
Δ = 102-4·(-2)·(-10)
Δ = 20
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:

$\sqrt{\Delta}=\sqrt{20}=\sqrt{4*5}=\sqrt{4}*\sqrt{5}=2\sqrt{5}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(10)-2\sqrt{5}}{2*-2}=\frac{-10-2\sqrt{5}}{-4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(10)+2\sqrt{5}}{2*-2}=\frac{-10+2\sqrt{5}}{-4}
$