1/2x-3=3-2x+4

Solution for 1/2x-3=3-2x+4 equation:

1/2x-3=3-2x+4

We move all terms to the left:

1/2x-3-(3-2x+4)=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+7)-3=0

We get rid of parentheses

1/2x+2x-7-3=0

We multiply all the terms by the denominator


2x*2x-7*2x-3*2x+1=0

Wy multiply elements

4x^2-14x-6x+1=0

We add all the numbers together, and all the variables

4x^2-20x+1=0

a = 4; b = -20; c = +1;
Δ = b2-4ac
Δ = -202-4·4·1
Δ = 384
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:

$\sqrt{\Delta}=\sqrt{384}=\sqrt{64*6}=\sqrt{64}*\sqrt{6}=8\sqrt{6}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-20)-8\sqrt{6}}{2*4}=\frac{20-8\sqrt{6}}{8}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-20)+8\sqrt{6}}{2*4}=\frac{20+8\sqrt{6}}{8}
$