10+x+1/2x+x+1/2x=18

Solution for 10+x+1/2x+x+1/2x=18 equation:

10+x+1/2x+x+1/2x=18

We move all terms to the left:

10+x+1/2x+x+1/2x-(18)=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+1/2x-8=0

We multiply all the terms by the denominator


2x*2x-8*2x+1+1=0

We add all the numbers together, and all the variables

2x*2x-8*2x+2=0

Wy multiply elements

4x^2-16x+2=0

a = 4; b = -16; c = +2;
Δ = b2-4ac
Δ = -162-4·4·2
Δ = 224
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:

$\sqrt{\Delta}=\sqrt{224}=\sqrt{16*14}=\sqrt{16}*\sqrt{14}=4\sqrt{14}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-16)-4\sqrt{14}}{2*4}=\frac{16-4\sqrt{14}}{8}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-16)+4\sqrt{14}}{2*4}=\frac{16+4\sqrt{14}}{8}
$