102-5/4n=3n

Solution for 102-5/4n=3n equation:

102-5/4n=3n

We move all terms to the left:

102-5/4n-(3n)=0


Domain of the equation: 4n!=0

n!=0/4

n!=0

n∈R


We add all the numbers together, and all the variables

-3n-5/4n+102=0

We multiply all the terms by the denominator


-3n*4n+102*4n-5=0

Wy multiply elements

-12n^2+408n-5=0

a = -12; b = 408; c = -5;
Δ = b2-4ac
Δ = 4082-4·(-12)·(-5)
Δ = 166224
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:

$\sqrt{\Delta}=\sqrt{166224}=\sqrt{16*10389}=\sqrt{16}*\sqrt{10389}=4\sqrt{10389}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(408)-4\sqrt{10389}}{2*-12}=\frac{-408-4\sqrt{10389}}{-24}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(408)+4\sqrt{10389}}{2*-12}=\frac{-408+4\sqrt{10389}}{-24}
$