x-6=3/2x

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

x-6=3/2x

We move all terms to the left:

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


Domain of the equation: 2x)!=0

x!=0/1

x!=0

x∈R


We add all the numbers together, and all the variables

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

We get rid of parentheses

x-3/2x-6=0

We multiply all the terms by the denominator


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

Wy multiply elements

2x^2-12x-3=0

a = 2; b = -12; c = -3;
Δ = b2-4ac
Δ = -122-4·2·(-3)
Δ = 168
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:

$\sqrt{\Delta}=\sqrt{168}=\sqrt{4*42}=\sqrt{4}*\sqrt{42}=2\sqrt{42}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-12)-2\sqrt{42}}{2*2}=\frac{12-2\sqrt{42}}{4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-12)+2\sqrt{42}}{2*2}=\frac{12+2\sqrt{42}}{4}
$