1/5n+3n=288

Solution for 1/5n+3n=288 equation:

1/5n+3n=288

We move all terms to the left:

1/5n+3n-(288)=0


Domain of the equation: 5n!=0

n!=0/5

n!=0

n∈R


We add all the numbers together, and all the variables

3n+1/5n-288=0

We multiply all the terms by the denominator


3n*5n-288*5n+1=0

Wy multiply elements

15n^2-1440n+1=0

a = 15; b = -1440; c = +1;
Δ = b2-4ac
Δ = -14402-4·15·1
Δ = 2073540
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:

$\sqrt{\Delta}=\sqrt{2073540}=\sqrt{4*518385}=\sqrt{4}*\sqrt{518385}=2\sqrt{518385}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-1440)-2\sqrt{518385}}{2*15}=\frac{1440-2\sqrt{518385}}{30}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-1440)+2\sqrt{518385}}{2*15}=\frac{1440+2\sqrt{518385}}{30}
$