1/2x-1=x+2

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

1/2x-1=x+2

We move all terms to the left:

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


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

1/2x-x-2-1=0

We multiply all the terms by the denominator


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

Wy multiply elements

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

We add all the numbers together, and all the variables

-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}
$