5/(x-3)+8/x=3

Solution for 5/(x-3)+8/x=3 equation:

5/(x-3)+8/x=3

We move all terms to the left:

5/(x-3)+8/x-(3)=0


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

We move all terms containing x to the left, all other terms to the right

x!=3

x∈R



Domain of the equation: x!=0

x∈R


We calculate fractions


5x/(x^2-3x)+(8x-24)/(x^2-3x)-3=0

We multiply all the terms by the denominator


5x+(8x-24)-3*(x^2-3x)=0

We multiply parentheses

-3x^2+5x+(8x-24)+9x=0

We get rid of parentheses

-3x^2+5x+8x+9x-24=0

We add all the numbers together, and all the variables

-3x^2+22x-24=0

a = -3; b = 22; c = -24;
Δ = b2-4ac
Δ = 222-4·(-3)·(-24)
Δ = 196
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{196}=14$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(22)-14}{2*-3}=\frac{-36}{-6}
=+6
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(22)+14}{2*-3}=\frac{-8}{-6}
=1+1/3
$