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

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

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

We move all terms to the left:

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


Domain of the equation: 2x+3)!=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}
$