5/x+1-4/2x+2+2/2x+1=13/18

Solution for 5/x+1-4/2x+2+2/2x+1=13/18 equation:

5/x+1-4/2x+2+2/2x+1=13/18

We move all terms to the left:

5/x+1-4/2x+2+2/2x+1-(13/18)=0


Domain of the equation: x!=0

x∈R



Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We add all the numbers together, and all the variables

5/x-4/2x+2/2x+1+2+1-(+13/18)=0

We add all the numbers together, and all the variables

5/x-4/2x+2/2x+4-(+13/18)=0

We get rid of parentheses

5/x-4/2x+2/2x+4-13/18=0

We calculate fractions


(-52x^2)/36x^2+180x/36x^2+(36x-4)/36x^2+4=0

We multiply all the terms by the denominator


(-52x^2)+180x+(36x-4)+4*36x^2=0

Wy multiply elements

(-52x^2)+144x^2+180x+(36x-4)=0

We get rid of parentheses

-52x^2+144x^2+180x+36x-4=0

We add all the numbers together, and all the variables

92x^2+216x-4=0

a = 92; b = 216; c = -4;
Δ = b2-4ac
Δ = 2162-4·92·(-4)
Δ = 48128
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:

$\sqrt{\Delta}=\sqrt{48128}=\sqrt{1024*47}=\sqrt{1024}*\sqrt{47}=32\sqrt{47}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(216)-32\sqrt{47}}{2*92}=\frac{-216-32\sqrt{47}}{184}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(216)+32\sqrt{47}}{2*92}=\frac{-216+32\sqrt{47}}{184}
$