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

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

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

We move all terms to the left:

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


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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

We add all the numbers together, and all the variables

-x*2x-3*2x-10*2x+4=0

Wy multiply elements

-2x^2-6x-20x+4=0

We add all the numbers together, and all the variables

-2x^2-26x+4=0

a = -2; b = -26; c = +4;
Δ = b2-4ac
Δ = -262-4·(-2)·4
Δ = 708
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:

$\sqrt{\Delta}=\sqrt{708}=\sqrt{4*177}=\sqrt{4}*\sqrt{177}=2\sqrt{177}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-26)-2\sqrt{177}}{2*-2}=\frac{26-2\sqrt{177}}{-4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-26)+2\sqrt{177}}{2*-2}=\frac{26+2\sqrt{177}}{-4}
$