4/9x+2=5/x-3

Solution for 4/9x+2=5/x-3 equation:

4/9x+2=5/x-3

We move all terms to the left:

4/9x+2-(5/x-3)=0


Domain of the equation: 9x!=0

x!=0/9

x!=0

x∈R



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

x∈R


We get rid of parentheses

4/9x-5/x+3+2=0

We calculate fractions


4x/9x^2+(-45x)/9x^2+3+2=0

We add all the numbers together, and all the variables

4x/9x^2+(-45x)/9x^2+5=0

We multiply all the terms by the denominator


4x+(-45x)+5*9x^2=0

Wy multiply elements

45x^2+4x+(-45x)=0

We get rid of parentheses

45x^2+4x-45x=0

We add all the numbers together, and all the variables

45x^2-41x=0

a = 45; b = -41; c = 0;
Δ = b2-4ac
Δ = -412-4·45·0
Δ = 1681
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{1681}=41$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-41)-41}{2*45}=\frac{0}{90}
=0
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-41)+41}{2*45}=\frac{82}{90}
=41/45
$