Proving a function is an inner product

[Vespero]

Homework Statement

I'm supposed to show that a function is an inner product if and only if b^2 - ac < 0 and a > 0.

I have proven all of the properties except that <x,x> > 0 if x!= 0. I would write the function out, but can't seem to get matrices to work.

The function is the product of a 1x2 matrix with entries x1 and x2, a 2x2 matrix with entries a, b, b, c, (filling in the top row, then moving to the bottom row), and a 2x1 matrix with entries y1, y2.

After multiplication, I arrive at
<x,x> = a(x1)^2 + 2b(x1)(x2) + c(x2)^2 > 0.

Now I must prove that this is true if and only if b^2 - ac < 0 and a > 0, but am not sure how to go about this.

Homework Equations

The Attempt at a Solution

I thought about different ways of inserting the conditions into the inequality or assuming that they were false and trying to arrive at a contradiction, but I can't seem to utilize them in a useful way.

 

[Hurkyl]

Some other things to try are:

Plugging in specific values for x1 and x2 to get a system of inequalities involving only a, b, and c.

Using the power of linear algebra to analyze the matrix and see if you can simply the problem or otherwise transform it into an easier one.

Using the power of high-school algebra to study and simplify a quadratic equation.

 

[jambaugh]

I'm guessing your function is:
$$\langle \mathbf{x},\mathbf{y}\rangle = \left( \begin{array}{c c} x_1 & x_2 \end{array}\right) \left( \begin{array}{c c} a & b\\ b & c\end{array}\right)\left(\begin{array}{c} y_1\\y_2 \end{array}\right)$$

I think your best bet is to complete the square on the quadratic. Treat it as a quadratic polynomial in x1.

Recall the process of completing the square:
$$x^2 + 2Bx+C = x^2 + 2Bx + B^2 - B^2 + C = (x+B)^2 -B^2 + C$$