1/2x-3+1=2x-20-3

Solution for 1/2x-3+1=2x-20-3 equation:

1/2x-3+1=2x-20-3

We move all terms to the left:

1/2x-3+1-(2x-20-3)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We add all the numbers together, and all the variables

1/2x-(2x-23)-3+1=0

We add all the numbers together, and all the variables

1/2x-(2x-23)-2=0

We get rid of parentheses

1/2x-2x+23-2=0

We multiply all the terms by the denominator


-2x*2x+23*2x-2*2x+1=0

Wy multiply elements

-4x^2+46x-4x+1=0

We add all the numbers together, and all the variables

-4x^2+42x+1=0

a = -4; b = 42; c = +1;
Δ = b2-4ac
Δ = 422-4·(-4)·1
Δ = 1780
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:

$\sqrt{\Delta}=\sqrt{1780}=\sqrt{4*445}=\sqrt{4}*\sqrt{445}=2\sqrt{445}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(42)-2\sqrt{445}}{2*-4}=\frac{-42-2\sqrt{445}}{-8}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(42)+2\sqrt{445}}{2*-4}=\frac{-42+2\sqrt{445}}{-8}
$