x+1/2x+x+200=165000

Solution for x+1/2x+x+200=165000 equation:

x+1/2x+x+200=165000

We move all terms to the left:

x+1/2x+x+200-(165000)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We add all the numbers together, and all the variables

2x+1/2x-164800=0

We multiply all the terms by the denominator


2x*2x-164800*2x+1=0

Wy multiply elements

4x^2-329600x+1=0

a = 4; b = -329600; c = +1;
Δ = b2-4ac
Δ = -3296002-4·4·1
Δ = 108636159984
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:

$\sqrt{\Delta}=\sqrt{108636159984}=\sqrt{1936*56113719}=\sqrt{1936}*\sqrt{56113719}=44\sqrt{56113719}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-329600)-44\sqrt{56113719}}{2*4}=\frac{329600-44\sqrt{56113719}}{8}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-329600)+44\sqrt{56113719}}{2*4}=\frac{329600+44\sqrt{56113719}}{8}
$