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

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

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

We move all terms to the left:

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


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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

We add all the numbers together, and all the variables

-x*2x-2*2x-10*2x+4=0

Wy multiply elements

-2x^2-4x-20x+4=0

We add all the numbers together, and all the variables

-2x^2-24x+4=0

a = -2; b = -24; c = +4;
Δ = b2-4ac
Δ = -242-4·(-2)·4
Δ = 608
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:

$\sqrt{\Delta}=\sqrt{608}=\sqrt{16*38}=\sqrt{16}*\sqrt{38}=4\sqrt{38}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-24)-4\sqrt{38}}{2*-2}=\frac{24-4\sqrt{38}}{-4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-24)+4\sqrt{38}}{2*-2}=\frac{24+4\sqrt{38}}{-4}
$