1/2x-3=4-x-22

Solution for 1/2x-3=4-x-22 equation:

1/2x-3=4-x-22

We move all terms to the left:

1/2x-3-(4-x-22)=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-(-1x-18)-3=0

We get rid of parentheses

1/2x+1x+18-3=0

We multiply all the terms by the denominator


1x*2x+18*2x-3*2x+1=0

Wy multiply elements

2x^2+36x-6x+1=0

We add all the numbers together, and all the variables

2x^2+30x+1=0

a = 2; b = 30; c = +1;
Δ = b2-4ac
Δ = 302-4·2·1
Δ = 892
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:

$\sqrt{\Delta}=\sqrt{892}=\sqrt{4*223}=\sqrt{4}*\sqrt{223}=2\sqrt{223}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(30)-2\sqrt{223}}{2*2}=\frac{-30-2\sqrt{223}}{4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(30)+2\sqrt{223}}{2*2}=\frac{-30+2\sqrt{223}}{4}
$