5+8n=50+1/2n

Solution for 5+8n=50+1/2n equation:

5+8n=50+1/2n

We move all terms to the left:

5+8n-(50+1/2n)=0


Domain of the equation: 2n)!=0

n!=0/1

n!=0

n∈R


We add all the numbers together, and all the variables

8n-(1/2n+50)+5=0

We get rid of parentheses

8n-1/2n-50+5=0

We multiply all the terms by the denominator


8n*2n-50*2n+5*2n-1=0

Wy multiply elements

16n^2-100n+10n-1=0

We add all the numbers together, and all the variables

16n^2-90n-1=0

a = 16; b = -90; c = -1;
Δ = b2-4ac
Δ = -902-4·16·(-1)
Δ = 8164
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:

$\sqrt{\Delta}=\sqrt{8164}=\sqrt{4*2041}=\sqrt{4}*\sqrt{2041}=2\sqrt{2041}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-90)-2\sqrt{2041}}{2*16}=\frac{90-2\sqrt{2041}}{32}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-90)+2\sqrt{2041}}{2*16}=\frac{90+2\sqrt{2041}}{32}
$