1/2x-5=x+3

Solution for 1/2x-5=x+3 equation:

1/2x-5=x+3

We move all terms to the left:

1/2x-5-(x+3)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

1/2x-x-3-5=0

We multiply all the terms by the denominator


-x*2x-3*2x-5*2x+1=0

Wy multiply elements

-2x^2-6x-10x+1=0

We add all the numbers together, and all the variables

-2x^2-16x+1=0

a = -2; b = -16; c = +1;
Δ = b2-4ac
Δ = -162-4·(-2)·1
Δ = 264
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:

$\sqrt{\Delta}=\sqrt{264}=\sqrt{4*66}=\sqrt{4}*\sqrt{66}=2\sqrt{66}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-16)-2\sqrt{66}}{2*-2}=\frac{16-2\sqrt{66}}{-4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-16)+2\sqrt{66}}{2*-2}=\frac{16+2\sqrt{66}}{-4}
$