4/5x+6-1/3x=2/5x+9

Solution for 4/5x+6-1/3x=2/5x+9 equation:

4/5x+6-1/3x=2/5x+9

We move all terms to the left:

4/5x+6-1/3x-(2/5x+9)=0


Domain of the equation: 5x!=0

x!=0/5

x!=0

x∈R



Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R



Domain of the equation: 5x+9)!=0

x∈R


We get rid of parentheses

4/5x-1/3x-2/5x-9+6=0

We calculate fractions


(-6x+4)/15x^2+(-5x)/15x^2-9+6=0

We add all the numbers together, and all the variables

(-6x+4)/15x^2+(-5x)/15x^2-3=0

We multiply all the terms by the denominator


(-6x+4)+(-5x)-3*15x^2=0

Wy multiply elements

-45x^2+(-6x+4)+(-5x)=0

We get rid of parentheses

-45x^2-6x-5x+4=0

We add all the numbers together, and all the variables

-45x^2-11x+4=0

a = -45; b = -11; c = +4;
Δ = b2-4ac
Δ = -112-4·(-45)·4
Δ = 841
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{841}=29$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-11)-29}{2*-45}=\frac{-18}{-90}
=1/5
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-11)+29}{2*-45}=\frac{40}{-90}
=-4/9
$