22/2+3n=4/5n+15

Solution for 22/2+3n=4/5n+15 equation:

22/2+3n=4/5n+15

We move all terms to the left:

22/2+3n-(4/5n+15)=0


Domain of the equation: 5n+15)!=0

n∈R


We add all the numbers together, and all the variables

3n-(4/5n+15)+11=0

We get rid of parentheses

3n-4/5n-15+11=0

We multiply all the terms by the denominator


3n*5n-15*5n+11*5n-4=0

Wy multiply elements

15n^2-75n+55n-4=0

We add all the numbers together, and all the variables

15n^2-20n-4=0

a = 15; b = -20; c = -4;
Δ = b2-4ac
Δ = -202-4·15·(-4)
Δ = 640
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{640}=\sqrt{64*10}=\sqrt{64}*\sqrt{10}=8\sqrt{10}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-20)-8\sqrt{10}}{2*15}=\frac{20-8\sqrt{10}}{30}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-20)+8\sqrt{10}}{2*15}=\frac{20+8\sqrt{10}}{30}
$