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

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

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

We move all terms to the left:

1/2x+2-3/2x-(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

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

We get rid of parentheses

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

We multiply all the terms by the denominator


-x*2x-19*2x+2*2x+1-3=0

We add all the numbers together, and all the variables

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

Wy multiply elements

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

We add all the numbers together, and all the variables

-2x^2-34x-2=0

a = -2; b = -34; c = -2;
Δ = b2-4ac
Δ = -342-4·(-2)·(-2)
Δ = 1140
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:

$\sqrt{\Delta}=\sqrt{1140}=\sqrt{4*285}=\sqrt{4}*\sqrt{285}=2\sqrt{285}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-34)-2\sqrt{285}}{2*-2}=\frac{34-2\sqrt{285}}{-4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-34)+2\sqrt{285}}{2*-2}=\frac{34+2\sqrt{285}}{-4}
$