3x-2+1/2x+x=21-2+x

Solution for 3x-2+1/2x+x=21-2+x equation:

3x-2+1/2x+x=21-2+x

We move all terms to the left:

3x-2+1/2x+x-(21-2+x)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We add all the numbers together, and all the variables

3x+1/2x+x-(x+19)-2=0

We add all the numbers together, and all the variables

4x+1/2x-(x+19)-2=0

We get rid of parentheses

4x+1/2x-x-19-2=0

We multiply all the terms by the denominator


4x*2x-x*2x-19*2x-2*2x+1=0

Wy multiply elements

8x^2-2x^2-38x-4x+1=0

We add all the numbers together, and all the variables

6x^2-42x+1=0

a = 6; b = -42; c = +1;
Δ = b2-4ac
Δ = -422-4·6·1
Δ = 1740
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:

$\sqrt{\Delta}=\sqrt{1740}=\sqrt{4*435}=\sqrt{4}*\sqrt{435}=2\sqrt{435}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-42)-2\sqrt{435}}{2*6}=\frac{42-2\sqrt{435}}{12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-42)+2\sqrt{435}}{2*6}=\frac{42+2\sqrt{435}}{12}
$