4x+1/5x-1=8x-1/5x+1

Solution for 4x+1/5x-1=8x-1/5x+1 equation:

4x+1/5x-1=8x-1/5x+1

We move all terms to the left:

4x+1/5x-1-(8x-1/5x+1)=0


Domain of the equation: 5x!=0

x!=0/5

x!=0

x∈R



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

x∈R


We get rid of parentheses

4x+1/5x-8x+1/5x-1-1=0

We multiply all the terms by the denominator


4x*5x-8x*5x-1*5x-1*5x+1+1=0

We add all the numbers together, and all the variables

4x*5x-8x*5x-1*5x-1*5x+2=0

Wy multiply elements

20x^2-40x^2-5x-5x+2=0

We add all the numbers together, and all the variables

-20x^2-10x+2=0

a = -20; b = -10; c = +2;
Δ = b2-4ac
Δ = -102-4·(-20)·2
Δ = 260
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:

$\sqrt{\Delta}=\sqrt{260}=\sqrt{4*65}=\sqrt{4}*\sqrt{65}=2\sqrt{65}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-10)-2\sqrt{65}}{2*-20}=\frac{10-2\sqrt{65}}{-40}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-10)+2\sqrt{65}}{2*-20}=\frac{10+2\sqrt{65}}{-40}
$