4/5x-16/3x=20/5x+32

Solution for 4/5x-16/3x=20/5x+32 equation:

4/5x-16/3x=20/5x+32

We move all terms to the left:

4/5x-16/3x-(20/5x+32)=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+32)!=0

x∈R


We get rid of parentheses

4/5x-16/3x-20/5x-32=0

We calculate fractions


(-60x+4)/15x^2+(-80x)/15x^2-32=0

We multiply all the terms by the denominator


(-60x+4)+(-80x)-32*15x^2=0

Wy multiply elements

-480x^2+(-60x+4)+(-80x)=0

We get rid of parentheses

-480x^2-60x-80x+4=0

We add all the numbers together, and all the variables

-480x^2-140x+4=0

a = -480; b = -140; c = +4;
Δ = b2-4ac
Δ = -1402-4·(-480)·4
Δ = 27280
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:

$\sqrt{\Delta}=\sqrt{27280}=\sqrt{16*1705}=\sqrt{16}*\sqrt{1705}=4\sqrt{1705}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-140)-4\sqrt{1705}}{2*-480}=\frac{140-4\sqrt{1705}}{-960}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-140)+4\sqrt{1705}}{2*-480}=\frac{140+4\sqrt{1705}}{-960}
$