1/2x+3=2x-5

Solution for 1/2x+3=2x-5 equation:

1/2x+3=2x-5

We move all terms to the left:

1/2x+3-(2x-5)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

1/2x-2x+5+3=0

We multiply all the terms by the denominator


-2x*2x+5*2x+3*2x+1=0

Wy multiply elements

-4x^2+10x+6x+1=0

We add all the numbers together, and all the variables

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

a = -4; b = 16; c = +1;
Δ = b2-4ac
Δ = 162-4·(-4)·1
Δ = 272
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:

$\sqrt{\Delta}=\sqrt{272}=\sqrt{16*17}=\sqrt{16}*\sqrt{17}=4\sqrt{17}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(16)-4\sqrt{17}}{2*-4}=\frac{-16-4\sqrt{17}}{-8}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(16)+4\sqrt{17}}{2*-4}=\frac{-16+4\sqrt{17}}{-8}
$