66+57n+1/2n=180

Solution for 66+57n+1/2n=180 equation:

66+57n+1/2n=180

We move all terms to the left:

66+57n+1/2n-(180)=0


Domain of the equation: 2n!=0

n!=0/2

n!=0

n∈R


We add all the numbers together, and all the variables

57n+1/2n-114=0

We multiply all the terms by the denominator


57n*2n-114*2n+1=0

Wy multiply elements

114n^2-228n+1=0

a = 114; b = -228; c = +1;
Δ = b2-4ac
Δ = -2282-4·114·1
Δ = 51528
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:

$\sqrt{\Delta}=\sqrt{51528}=\sqrt{4*12882}=\sqrt{4}*\sqrt{12882}=2\sqrt{12882}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-228)-2\sqrt{12882}}{2*114}=\frac{228-2\sqrt{12882}}{228}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-228)+2\sqrt{12882}}{2*114}=\frac{228+2\sqrt{12882}}{228}
$