Solutions of quadratic equation with $n$ variables

Question

I'm trying to find the roots of a quadratic equation with $n$ variables. I've looked through the internet but I wasn't able to find any convincing formula.

Given a vector $v=\{x_1, x_2, x_3,\ldots, x_n\}$ with $n$ variables, I can rearrange the equation to the following form: $vAv^{\top}$. I've been working this out, but I was only able to find a general expression to see if this equation has any solutions. I was wondering if there exist a general formula for this type of problem.

Thanks in advance

Answer 1

Plug in your favourite values for $x_1,\ldots,x_{n-1}$. Now you have a quadratic equation in $x_n$, which you know how to solve. As you can tell, for $n>1$ there are infinitely many roots. (Provided that you're working over an algebraically closed field like the complex numbers).