1/5n+3n=2n+42

Solution for 1/5n+3n=2n+42 equation:

1/5n+3n=2n+42

We move all terms to the left:

1/5n+3n-(2n+42)=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-(2n+42)=0

We get rid of parentheses

3n+1/5n-2n-42=0

We multiply all the terms by the denominator


3n*5n-2n*5n-42*5n+1=0

Wy multiply elements

15n^2-10n^2-210n+1=0

We add all the numbers together, and all the variables

5n^2-210n+1=0

a = 5; b = -210; c = +1;
Δ = b2-4ac
Δ = -2102-4·5·1
Δ = 44080
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:

$\sqrt{\Delta}=\sqrt{44080}=\sqrt{16*2755}=\sqrt{16}*\sqrt{2755}=4\sqrt{2755}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-210)-4\sqrt{2755}}{2*5}=\frac{210-4\sqrt{2755}}{10}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-210)+4\sqrt{2755}}{2*5}=\frac{210+4\sqrt{2755}}{10}
$