3n+5/6n=46

Solution for 3n+5/6n=46 equation:

3n+5/6n=46

We move all terms to the left:

3n+5/6n-(46)=0


Domain of the equation: 6n!=0

n!=0/6

n!=0

n∈R


We multiply all the terms by the denominator


3n*6n-46*6n+5=0

Wy multiply elements

18n^2-276n+5=0

a = 18; b = -276; c = +5;
Δ = b2-4ac
Δ = -2762-4·18·5
Δ = 75816
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:

$\sqrt{\Delta}=\sqrt{75816}=\sqrt{2916*26}=\sqrt{2916}*\sqrt{26}=54\sqrt{26}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-276)-54\sqrt{26}}{2*18}=\frac{276-54\sqrt{26}}{36}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-276)+54\sqrt{26}}{2*18}=\frac{276+54\sqrt{26}}{36}
$