8=1/2n+8,n=

Solution for 8=1/2n+8,n= equation:

8=1/2n+8.n=

We move all terms to the left:

8-(1/2n+8.n)=0


Domain of the equation: 2n+8.n)!=0

n∈R


We add all the numbers together, and all the variables

-(+8.n+1/2n)+8=0

We get rid of parentheses

-8.n-1/2n+8=0

We multiply all the terms by the denominator


-(8.n)*2n+8*2n-1=0

We add all the numbers together, and all the variables

-(+8.n)*2n+8*2n-1=0

We multiply parentheses

-16n^2+8*2n-1=0

Wy multiply elements

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

a = -16; b = 16; c = -1;
Δ = b2-4ac
Δ = 162-4·(-16)·(-1)
Δ = 192
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:

$\sqrt{\Delta}=\sqrt{192}=\sqrt{64*3}=\sqrt{64}*\sqrt{3}=8\sqrt{3}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(16)-8\sqrt{3}}{2*-16}=\frac{-16-8\sqrt{3}}{-32}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(16)+8\sqrt{3}}{2*-16}=\frac{-16+8\sqrt{3}}{-32}
$