1/x+1/2x=4/x+2

Solution for 1/x+1/2x=4/x+2 equation:

1/x+1/2x=4/x+2

We move all terms to the left:

1/x+1/2x-(4/x+2)=0


Domain of the equation: x!=0

x∈R



Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R



Domain of the equation: x+2)!=0

x∈R


We get rid of parentheses

1/x+1/2x-4/x-2=0

We calculate fractions


(-8x+1)/2x^2+x/2x^2-2=0

We multiply all the terms by the denominator


(-8x+1)+x-2*2x^2=0

We add all the numbers together, and all the variables

x+(-8x+1)-2*2x^2=0

Wy multiply elements

-4x^2+x+(-8x+1)=0

We get rid of parentheses

-4x^2+x-8x+1=0

We add all the numbers together, and all the variables

-4x^2-7x+1=0

a = -4; b = -7; c = +1;
Δ = b2-4ac
Δ = -72-4·(-4)·1
Δ = 65
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-7)-\sqrt{65}}{2*-4}=\frac{7-\sqrt{65}}{-8}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-7)+\sqrt{65}}{2*-4}=\frac{7+\sqrt{65}}{-8}
$