1/2x+3=2x-8

Solution for 1/2x+3=2x-8 equation:

1/2x+3=2x-8

We move all terms to the left:

1/2x+3-(2x-8)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

1/2x-2x+8+3=0

We multiply all the terms by the denominator


-2x*2x+8*2x+3*2x+1=0

Wy multiply elements

-4x^2+16x+6x+1=0

We add all the numbers together, and all the variables

-4x^2+22x+1=0

a = -4; b = 22; c = +1;
Δ = b2-4ac
Δ = 222-4·(-4)·1
Δ = 500
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:

$\sqrt{\Delta}=\sqrt{500}=\sqrt{100*5}=\sqrt{100}*\sqrt{5}=10\sqrt{5}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(22)-10\sqrt{5}}{2*-4}=\frac{-22-10\sqrt{5}}{-8}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(22)+10\sqrt{5}}{2*-4}=\frac{-22+10\sqrt{5}}{-8}
$