x+1/2x=111

Solution for x+1/2x=111 equation:

x+1/2x=111

We move all terms to the left:

x+1/2x-(111)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We multiply all the terms by the denominator


x*2x-111*2x+1=0

Wy multiply elements

2x^2-222x+1=0

a = 2; b = -222; c = +1;
Δ = b2-4ac
Δ = -2222-4·2·1
Δ = 49276
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:

$\sqrt{\Delta}=\sqrt{49276}=\sqrt{4*12319}=\sqrt{4}*\sqrt{12319}=2\sqrt{12319}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-222)-2\sqrt{12319}}{2*2}=\frac{222-2\sqrt{12319}}{4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-222)+2\sqrt{12319}}{2*2}=\frac{222+2\sqrt{12319}}{4}
$