1/2x+3=23x+1

Solution for 1/2x+3=23x+1 equation:

1/2x+3=23x+1

We move all terms to the left:

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


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


-23x*2x-1*2x+3*2x+1=0

Wy multiply elements

-46x^2-2x+6x+1=0

We add all the numbers together, and all the variables

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

a = -46; b = 4; c = +1;
Δ = b2-4ac
Δ = 42-4·(-46)·1
Δ = 200
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:

$\sqrt{\Delta}=\sqrt{200}=\sqrt{100*2}=\sqrt{100}*\sqrt{2}=10\sqrt{2}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(4)-10\sqrt{2}}{2*-46}=\frac{-4-10\sqrt{2}}{-92}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(4)+10\sqrt{2}}{2*-46}=\frac{-4+10\sqrt{2}}{-92}
$