4^-2n/16^-1=4^n+1

Solution for 4^-2n/16^-1=4^n+1 equation:

4^-2n/16^-1=4^n+1

We move all terms to the left:

4^-2n/16^-1-(4^n+1)=0

We add all the numbers together, and all the variables

-2n/16^-(4^n+1)=0

We get rid of parentheses

-2n/16^-4^n-1=0

We multiply all the terms by the denominator


-2n-4^n*16^-1*16^=0

We add all the numbers together, and all the variables

-2n-4^n*16^=0

Wy multiply elements

-64n^2-2n=0

a = -64; b = -2; c = 0;
Δ = b2-4ac
Δ = -22-4·(-64)·0
Δ = 4
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{4}=2$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-2)-2}{2*-64}=\frac{0}{-128}
=0
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-2)+2}{2*-64}=\frac{4}{-128}
=-1/32
$