4/x-5-1/3x-5=-8/3x-15

Solution for 4/x-5-1/3x-5=-8/3x-15 equation:

4/x-5-1/3x-5=-8/3x-15

We move all terms to the left:

4/x-5-1/3x-5-(-8/3x-15)=0


Domain of the equation: x!=0

x∈R



Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R



Domain of the equation: 3x-15)!=0

x∈R


We add all the numbers together, and all the variables

4/x-1/3x-(-8/3x-15)-10=0

We get rid of parentheses

4/x-1/3x+8/3x+15-10=0

We calculate fractions


12x/3x^2+(8x-1)/3x^2+15-10=0

We add all the numbers together, and all the variables

12x/3x^2+(8x-1)/3x^2+5=0

We multiply all the terms by the denominator


12x+(8x-1)+5*3x^2=0

Wy multiply elements

15x^2+12x+(8x-1)=0

We get rid of parentheses

15x^2+12x+8x-1=0

We add all the numbers together, and all the variables

15x^2+20x-1=0

a = 15; b = 20; c = -1;
Δ = b2-4ac
Δ = 202-4·15·(-1)
Δ = 460
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:

$\sqrt{\Delta}=\sqrt{460}=\sqrt{4*115}=\sqrt{4}*\sqrt{115}=2\sqrt{115}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(20)-2\sqrt{115}}{2*15}=\frac{-20-2\sqrt{115}}{30}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(20)+2\sqrt{115}}{2*15}=\frac{-20+2\sqrt{115}}{30}
$