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n+1/3n=24
We move all terms to the left:
n+1/3n-(24)=0
Domain of the equation: 3n!=0We multiply all the terms by the denominator
n!=0/3
n!=0
n∈R
n*3n-24*3n+1=0
Wy multiply elements
3n^2-72n+1=0
a = 3; b = -72; c = +1;
Δ = b2-4ac
Δ = -722-4·3·1
Δ = 5172
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}$$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$
The end solution:
$\sqrt{\Delta}=\sqrt{5172}=\sqrt{4*1293}=\sqrt{4}*\sqrt{1293}=2\sqrt{1293}$$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-72)-2\sqrt{1293}}{2*3}=\frac{72-2\sqrt{1293}}{6} $$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-72)+2\sqrt{1293}}{2*3}=\frac{72+2\sqrt{1293}}{6} $
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