-1/2x-3=23+x

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

-1/2x-3=23+x

We move all terms to the left:

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

We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

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

We add all the numbers together, and all the variables

-2x^2-52x-1=0

a = -2; b = -52; c = -1;
Δ = b2-4ac
Δ = -522-4·(-2)·(-1)
Δ = 2696
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:

$\sqrt{\Delta}=\sqrt{2696}=\sqrt{4*674}=\sqrt{4}*\sqrt{674}=2\sqrt{674}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-52)-2\sqrt{674}}{2*-2}=\frac{52-2\sqrt{674}}{-4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-52)+2\sqrt{674}}{2*-2}=\frac{52+2\sqrt{674}}{-4}
$