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## 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.