Solving characteristic equations...

Question

Can someone explain to me what my teacher is doing? $x^2 - ax - b = 0$ ..? Isn;t he using the quadratic formula to solve this problem? If that's the case, then where is the $c$ at? Shouldn't he have said $ax^2 - bx - c = 0$ How do we know what $c$ is? Also how do we know that $a=b=1$ ?? This is for a discrete math class and I am so confused. Any guidance would be great.

Answer 1

The letters $a,b$ and $c$ are just letters. I think your problem is with the quadratic formula itself. Think like this: $$\color{red}{\rm stuff}\,x^2 + \color{blue}{\rm stuff}\,x + \color{green}{\rm stuff} = 0 \implies x = \frac{-\color{blue}{\rm stuff} \pm \sqrt{(\color{blue}{\rm stuff})^2 - 4\,\color{red}{\rm stuff}\,\color{green}{\rm stuff}}}{2\,\color{red}{\rm stuff}}$$

Don't think of letters, think of postitions (I used colors above to try to ilustrate that). For example: $$bx^2+cx+a = 0 \implies x = \frac{-c\pm\sqrt{c^2-4ab}}{2b},$$etc.

Answer 2

The $c$ doesn't have to do anything with the quadratic.

We know that $s_n=s_{n-1}+s_{n-2}$, thus $s_n-s_{n-1}-s_{n-2}=0$. Now replace $s_n$ with $x^2$, $s_{n-1}$ with $x$ and $s_{n-2}$ with 1 to get the characteristic equation.