55n=5/6n+5

Solution for 55n=5/6n+5 equation:

55n=5/6n+5

We move all terms to the left:

55n-(5/6n+5)=0


Domain of the equation: 6n+5)!=0

n∈R


We get rid of parentheses

55n-5/6n-5=0

We multiply all the terms by the denominator


55n*6n-5*6n-5=0

Wy multiply elements

330n^2-30n-5=0

a = 330; b = -30; c = -5;
Δ = b2-4ac
Δ = -302-4·330·(-5)
Δ = 7500
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:

$\sqrt{\Delta}=\sqrt{7500}=\sqrt{2500*3}=\sqrt{2500}*\sqrt{3}=50\sqrt{3}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-30)-50\sqrt{3}}{2*330}=\frac{30-50\sqrt{3}}{660}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-30)+50\sqrt{3}}{2*330}=\frac{30+50\sqrt{3}}{660}
$