4x+5/5=4/x+4-1

Solution for 4x+5/5=4/x+4-1 equation:

4x+5/5=4/x+4-1

We move all terms to the left:

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


Domain of the equation: x+4-1)!=0

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

x-1)!=-4

x∈R


We add all the numbers together, and all the variables

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

We add all the numbers together, and all the variables

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

We get rid of parentheses

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

We multiply all the terms by the denominator


4x*x-3*x+1*x-4=0

We add all the numbers together, and all the variables

-2x+4x*x-4=0

Wy multiply elements

4x^2-2x-4=0

a = 4; b = -2; c = -4;
Δ = b2-4ac
Δ = -22-4·4·(-4)
Δ = 68
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:

$\sqrt{\Delta}=\sqrt{68}=\sqrt{4*17}=\sqrt{4}*\sqrt{17}=2\sqrt{17}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-2)-2\sqrt{17}}{2*4}=\frac{2-2\sqrt{17}}{8}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-2)+2\sqrt{17}}{2*4}=\frac{2+2\sqrt{17}}{8}
$