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

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

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

We move all terms to the left:

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


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R



Domain of the equation: 5x-1)!=0

x∈R


We get rid of parentheses

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

We calculate fractions


125x/375x^2+(-3x)/375x^2+(3x+1)/375x^2+1=0

We multiply all the terms by the denominator


125x+(-3x)+(3x+1)+1*375x^2=0

Wy multiply elements

375x^2+125x+(-3x)+(3x+1)=0

We get rid of parentheses

375x^2+125x-3x+3x+1=0

We add all the numbers together, and all the variables

375x^2+125x+1=0

a = 375; b = 125; c = +1;
Δ = b2-4ac
Δ = 1252-4·375·1
Δ = 14125
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:

$\sqrt{\Delta}=\sqrt{14125}=\sqrt{25*565}=\sqrt{25}*\sqrt{565}=5\sqrt{565}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(125)-5\sqrt{565}}{2*375}=\frac{-125-5\sqrt{565}}{750}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(125)+5\sqrt{565}}{2*375}=\frac{-125+5\sqrt{565}}{750}
$