Finding x for a certain quadratic form

[vellum93]

Homework Statement

Suppose a,c>0 and $b^2-4ac>0$. Explain how you could find $x_1, x_2 ε ℝ$
such that $a(x_1) ^2+bx_1x_2+c(x_2)^2<0$.

Homework Equations

$$q\begin{pmatrix}x_1\\x_2\end{pmatrix} = a(x_1)^2+bx_1x_2+c(x_2)^2$$

The Attempt at a Solution

I'm not sure where to go with this. This is part (b) of a question and my answer for part (a) shows that q takes both positive and negative values. I also know this equation can be changed into a real symmetric matrix. Maybe I can use that or have q( x1 x2 ) equal the symmetric matrix?

 

[vellum93]

I think I've solved it. It involves setting up the equation q( x1 x2 x3) = [x1 x2 x3] * A * (x1 x2)