# Solutions of quadratic equation with n variables

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 $v=${$x_1, x_2, x_3, ..., x_n$} with $n$ variables, I can rearrange the equation to the following form: $vAv^{T}$. 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.

• what equation? what type of entries does $A$ have? what type of entries can $v$ have? – Will Jagy Jun 4 '16 at 19:49
• For example: $ax^2+bxy+cy^2+dx+ey+fxz+h=0$. I'm looking for an expression like the quadratic form for one variable, only for multiple variables. I've found that the $det(A)<0$ will tell me whether the equation will have multiple solutions. – Curious Jun 4 '16 at 19:57
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).