102-5/4n=3n

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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} $

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