In the book the quadratic form associated with a bilinear form f is defined as $q(\alpha)=f(\alpha,\alpha)$. Then, if $U$ is a linear operator on $\mathbb R^2$ an operator $U^\dagger$ on the space of quadratic forms on $\mathbb R^2$ is defined by the equation $U^\dagger q(\alpha)=q(U(\alpha))$. Then, an exercise asks to find an operator $U$ such that if the quadratic form $q$ on $\mathbb R^2$ is $q=ax_1^2+2bx_1x_2+cx_2^2$, then $U^\dagger q=ax_1^2+\left(c- \frac {b^2}a\right)x_2^2$
A hint is given that recommends to complete the square in other to find $U^{-1}$, but I have no idea what to do.
I've already tried several things but it is sadly clear I'm not understanding the hint given at all. This is what I've managed to do so far (too long for a comment):
Completing the square in the unique place where it makes some sense to me:
$$q(x,y):=ax^2+2bxy+cy^2=a\left(x+\frac bay\right)^2+\left(c-\frac{b^2}a\right)y^2$$
Now, any linear operator on the real plane is of the form
$$U\binom xy=\binom{\alpha x+\beta y}{\gamma x+\delta y}\;,\;\;\alpha,\beta,\gamma,\delta\in\Bbb R$$
(This is a good exercise: a map $\;T:\Bbb F^n\to\Bbb F^m\;,\;\;\Bbb F\;$ a field, is a linear map iff it is of the form
$$T(x_1,\ldots,x_n):=\left(t_1(x_1,...,x_n),\ldots,t_m(x_1,..,x_n)\right)$$
with every $\;t_i\;$ a homogeneous polynomial of degree one in $\;x_1,...,x_n\;$) .
So we then have
$$U^\dagger(q(x,y))=q\left(U\binom xy\right):=a(\alpha x+\beta y)^2+2b(\alpha x+\beta y)(\gamma x+\delta y)+c(\gamma x+\delta y)^2$$
Somehow then, it seems to be we should be able to compare (equality?) the last xpression with the first one above and get what the coefficients $\;\alpha,\beta,\gamma,\delta\;$ are...
Where and how does $\;U^{-1}\;$ kick in with the "complete the square" thing is, so far, beyond my comprehension...