x=1/2x-3

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

x=1/2x-3

We move all terms to the left:

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


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

x∈R


We get rid of parentheses

x-1/2x+3=0

We multiply all the terms by the denominator


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

Wy multiply elements

2x^2+6x-1=0

a = 2; b = 6; c = -1;
Δ = b2-4ac
Δ = 62-4·2·(-1)
Δ = 44
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:

$\sqrt{\Delta}=\sqrt{44}=\sqrt{4*11}=\sqrt{4}*\sqrt{11}=2\sqrt{11}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(6)-2\sqrt{11}}{2*2}=\frac{-6-2\sqrt{11}}{4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(6)+2\sqrt{11}}{2*2}=\frac{-6+2\sqrt{11}}{4}
$