10-5x/3x-2/x+5=8-4x/x+5

Solution for 10-5x/3x-2/x+5=8-4x/x+5 equation:

10-5x/3x-2/x+5=8-4x/x+5

We move all terms to the left:

10-5x/3x-2/x+5-(8-4x/x+5)=0


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R



Domain of the equation: x!=0

x∈R



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

x∈R


We add all the numbers together, and all the variables

-5x/3x-2/x-(-4x/x+13)+10+5=0

We add all the numbers together, and all the variables

-5x/3x-2/x-(-4x/x+13)+15=0

We get rid of parentheses

-5x/3x-2/x+4x/x-13+15=0

We calculate fractions


(-5x^2)/3x^2+(12x^2-2)/3x^2-13+15=0

We add all the numbers together, and all the variables

(-5x^2)/3x^2+(12x^2-2)/3x^2+2=0

We multiply all the terms by the denominator


(-5x^2)+(12x^2-2)+2*3x^2=0

Wy multiply elements

(-5x^2)+6x^2+(12x^2-2)=0

We get rid of parentheses

-5x^2+6x^2+12x^2-2=0

We add all the numbers together, and all the variables

13x^2-2=0

a = 13; b = 0; c = -2;
Δ = b2-4ac
Δ = 02-4·13·(-2)
Δ = 104
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:

$\sqrt{\Delta}=\sqrt{104}=\sqrt{4*26}=\sqrt{4}*\sqrt{26}=2\sqrt{26}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-2\sqrt{26}}{2*13}=\frac{0-2\sqrt{26}}{26}
=-\frac{2\sqrt{26}}{26}
=-\frac{\sqrt{26}}{13}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+2\sqrt{26}}{2*13}=\frac{0+2\sqrt{26}}{26}
=\frac{2\sqrt{26}}{26}
=\frac{\sqrt{26}}{13}
$