5y-1/y+4=y+9/y+4

Solution for 5y-1/y+4=y+9/y+4 equation:

5y-1/y+4=y+9/y+4

We move all terms to the left:

5y-1/y+4-(y+9/y+4)=0


Domain of the equation: y!=0

y∈R



Domain of the equation: y+4)!=0

y∈R


We get rid of parentheses

5y-1/y-y-9/y-4+4=0

We multiply all the terms by the denominator


5y*y-y*y-4*y+4*y-1-9=0

We add all the numbers together, and all the variables

5y*y-y*y-10=0

Wy multiply elements

5y^2-1y^2-10=0

We add all the numbers together, and all the variables

4y^2-10=0

a = 4; b = 0; c = -10;
Δ = b2-4ac
Δ = 02-4·4·(-10)
Δ = 160
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{160}=\sqrt{16*10}=\sqrt{16}*\sqrt{10}=4\sqrt{10}$
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-4\sqrt{10}}{2*4}=\frac{0-4\sqrt{10}}{8}
=-\frac{4\sqrt{10}}{8}
=-\frac{\sqrt{10}}{2}
$
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+4\sqrt{10}}{2*4}=\frac{0+4\sqrt{10}}{8}
=\frac{4\sqrt{10}}{8}
=\frac{\sqrt{10}}{2}
$