Question

Can someone help me understand the solutions to this homogeneous equation?

Answer 1

\[\alpha ^{n}-5\alpha ^{n-1}+6\alpha^{n-2}=0\]

Answer 2

The solutions for alpha is 2 and 3, how come?

Answer 3

\[a^n-5a^{n-1}+6a^{n-2}=0\]
\[a^n-5a^{n-1}+6a^{n-2}=a^{n-2}(a^2-5a+6)=0\]
\[a^n \neq 0\]
that means \[a^2-5a+6=0\]
solve for a

Answer 4

Oh nice! that make sense but why is it not negative though?

Answer 5

because if you actually factor the quadratic equation and solve for a, the answers will be positive...

Answer 6

or because for negative numbers, even and odd roots will have different signs making it much more difficult for an actual solution for a, if any