3x/x-4+4=8-5x/4-x

Solution for 3x/x-4+4=8-5x/4-x equation:

3x/x-4+4=8-5x/4-x

We move all terms to the left:

3x/x-4+4-(8-5x/4-x)=0


Domain of the equation: x!=0

x∈R



Domain of the equation: 4-x)!=0

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

-x)!=-4

x!=-4/1

x!=-4

x∈R


We add all the numbers together, and all the variables

3x/x-(-1x-5x/4+8)-4+4=0

We add all the numbers together, and all the variables

3x/x-(-1x-5x/4+8)=0

We get rid of parentheses

3x/x+1x+5x/4-8=0

We calculate fractions


5x^2/4x+1x+12x/4x-8=0

We add all the numbers together, and all the variables

5x^2/4x+x+12x/4x-8=0

We multiply all the terms by the denominator


5x^2+x*4x+12x-8*4x=0

We add all the numbers together, and all the variables

5x^2+12x+x*4x-8*4x=0

Wy multiply elements

5x^2+4x^2+12x-32x=0

We add all the numbers together, and all the variables

9x^2-20x=0

a = 9; b = -20; c = 0;
Δ = b2-4ac
Δ = -202-4·9·0
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*9}=\frac{0}{18}
=0
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-20)+20}{2*9}=\frac{40}{18}
=2+2/9
$