1/x-1/x-5=1/3x

Solution for 1/x-1/x-5=1/3x equation:

1/x-1/x-5=1/3x

We move all terms to the left:

1/x-1/x-5-(1/3x)=0


Domain of the equation: x!=0

x∈R



Domain of the equation: 3x)!=0

x!=0/1

x!=0

x∈R


We add all the numbers together, and all the variables

1/x-1/x-(+1/3x)-5=0

We get rid of parentheses

1/x-1/x-1/3x-5=0

We calculate fractions


(-3x+1)/3x^2+(-x)/3x^2-5=0

We add all the numbers together, and all the variables

(-3x+1)/3x^2+(-1x)/3x^2-5=0

We multiply all the terms by the denominator


(-3x+1)+(-1x)-5*3x^2=0

Wy multiply elements

-15x^2+(-3x+1)+(-1x)=0

We get rid of parentheses

-15x^2-3x-1x+1=0

We add all the numbers together, and all the variables

-15x^2-4x+1=0

a = -15; b = -4; c = +1;
Δ = b2-4ac
Δ = -42-4·(-15)·1
Δ = 76
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:

$\sqrt{\Delta}=\sqrt{76}=\sqrt{4*19}=\sqrt{4}*\sqrt{19}=2\sqrt{19}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-4)-2\sqrt{19}}{2*-15}=\frac{4-2\sqrt{19}}{-30}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-4)+2\sqrt{19}}{2*-15}=\frac{4+2\sqrt{19}}{-30}
$