7/3x+2=9/x-6

Solution for 7/3x+2=9/x-6 equation:

7/3x+2=9/x-6

We move all terms to the left:

7/3x+2-(9/x-6)=0


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R



Domain of the equation: x-6)!=0

x∈R


We get rid of parentheses

7/3x-9/x+6+2=0

We calculate fractions


7x/3x^2+(-27x)/3x^2+6+2=0

We add all the numbers together, and all the variables

7x/3x^2+(-27x)/3x^2+8=0

We multiply all the terms by the denominator


7x+(-27x)+8*3x^2=0

Wy multiply elements

24x^2+7x+(-27x)=0

We get rid of parentheses

24x^2+7x-27x=0

We add all the numbers together, and all the variables

24x^2-20x=0

a = 24; b = -20; c = 0;
Δ = b2-4ac
Δ = -202-4·24·0
Δ = 400
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{400}=20$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-20)-20}{2*24}=\frac{0}{48}
=0
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-20)+20}{2*24}=\frac{40}{48}
=5/6
$