x-6=1/2x+3

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

x-6=1/2x+3

We move all terms to the left:

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


Domain of the equation: 2x+3)!=0

x∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

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

We add all the numbers together, and all the variables

2x^2-18x-1=0

a = 2; b = -18; c = -1;
Δ = b2-4ac
Δ = -182-4·2·(-1)
Δ = 332
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:

$\sqrt{\Delta}=\sqrt{332}=\sqrt{4*83}=\sqrt{4}*\sqrt{83}=2\sqrt{83}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-18)-2\sqrt{83}}{2*2}=\frac{18-2\sqrt{83}}{4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-18)+2\sqrt{83}}{2*2}=\frac{18+2\sqrt{83}}{4}
$