1/2x-6=2x+3

Solution for 1/2x-6=2x+3 equation:

1/2x-6=2x+3

We move all terms to the left:

1/2x-6-(2x+3)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

1/2x-2x-3-6=0

We multiply all the terms by the denominator


-2x*2x-3*2x-6*2x+1=0

Wy multiply elements

-4x^2-6x-12x+1=0

We add all the numbers together, and all the variables

-4x^2-18x+1=0

a = -4; b = -18; c = +1;
Δ = b2-4ac
Δ = -182-4·(-4)·1
Δ = 340
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:

$\sqrt{\Delta}=\sqrt{340}=\sqrt{4*85}=\sqrt{4}*\sqrt{85}=2\sqrt{85}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-18)-2\sqrt{85}}{2*-4}=\frac{18-2\sqrt{85}}{-8}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-18)+2\sqrt{85}}{2*-4}=\frac{18+2\sqrt{85}}{-8}
$