considering an equality with a polynomial of second degree where the coefficient for $x^2$ is $1$ I know that
$$ a x^2 + b x + c = a(x-\alpha)(x-\beta) = 0 $$
I also know that
$$ \alpha + \beta = -b/a $$ $$ \alpha \times \beta = c/a $$
At this point I'm with some pen and paper and I'm trying to solve such problem ( finding $\alpha$ and $\beta$ ) using a linear system, and I'm looking for help .
If I think about this as a matrix, for me the way to describe the last 2 equalities involving $\alpha$ and $\beta$ is : $$ \begin{matrix} \alpha & \beta \\ \beta & 0 \\ \end{matrix} $$
and the $Ax=b$ system should look like ( I'm still a newbie, bear with me, I'm trying to derive a linear system for the first time )
$$ \begin{pmatrix} \alpha & \beta \\ \beta & 0 \\ \end{pmatrix} \times \begin{pmatrix} ? \\ ? \\ \end{pmatrix} = \begin{pmatrix} -b/a \\ c/a \\ \end{pmatrix} $$
At this point I have 2 major problems :
- I have no idea what to put inside the "x" column vector for the $Ax=b$ problem
- $\alpha$ and $\beta$ are not that useful in that position since in input I only have $a,b$ and $c$ ; I think I should rework this formulation and I don't know how .
I think that I'm on the right track because for a second degree polynomial I have a $2 \times 2$ matrix, I'm having troubles formulating the right linear system for this problem by soving the 2 problems listed above .
EDIT 1
After the observation that the "x" column vector should be something like $\begin{pmatrix} \alpha \\ \beta \end{pmatrix}$ and that the division makes this a non-linear system, I tried to rework this further but I think that I need more help .
The only thing I remember that resembles a non linear behaviour simulated by a linear system / matrix , is an homogenous matrix, for example projective geometrive is non linear but computer aided rendering often times uses homogenous matrice / coordinates to do the job .
If this is the case I'm dead in the water because I never understood how this matrices really work and how to derive them .
The alternative is to exploit the fact that everything is divided by $a$ and so I can rewrite the "b" column vector by repositioning $a$ as a factor of the elements of the matrix . But I'm still confused about this because it will just complicate things .
Can you help me deploying this homogenous system ?