Simplifying with wxMaxima

Question

I have this expression in wxMaxima:

A:a[0] + sum(b[k]^2*a[k],k,1,n);
B:-2*sum((x-c[k]-b[k]*d[k])*b[k]*a[k],k,1,n);
C:sum((x-c[k]-b[k]*d[k])^2*a[k],k,1,n);
P:C-(B^2)/(4*A);

So, this is P:

$\left( \sum_{k=1}^{n}{a}_{k}\,{\left( x−{b}_{k}\,{d}_{k}−{c}_{k}\right) }^{2}\right) -\frac{{\left( \sum_{k=1}^{n}{a}_{k}\,{b}_{k}\,\left( x−{b}_{k}\,{d}_{k}−{c}_{k}\right) \right) }^{2}}{\left( \sum_{k=1}^{n}{a}_{k}\,{b}_{k}^{2}\right) +{a}_{0}}$

I want to express P as a second degree polynomial of x. But I cannot find the way with wxMaxima (neither with Sage).

Can you help me, please?

Answer 1

Hint

Let us consider your expression $$P=\left( \sum_{k=1}^{n}{a}_{k}\,{\left( x-{b}_{k}\,{d}_{k}-{c}_{k}\right) }^{2}\right) -\frac{{\left( \sum_{k=1}^{n}{a}_{k}\,{b}_{k}\,\left( x-{b}_{k}\,{d}_{k}-{c}_{k}\right) \right) }^{2}}{\left( \sum_{k=1}^{n}{a}_{k}\,{b}_{k}^{2}\right) +{a}_{0}}$$ Now, let us define some quantities as, for example, $$\frac{1}{\alpha}=\left(\sum_{k=1}^{n}{a}_{k}\,{b}_{k}^{2}\right) +{a}_{0}$$ $$u_k={b}_{k}\,{d}_{k}+{c}_{k}$$ $$v_k={a}_{k}\,{b}_{k}$$ So, replacing $$P=\sum_{k=1}^n a_k (x-u_k)^2-\alpha \left(\sum_{k=1}^n v_k(x-u_k)\right)^2=A+Bx+C x^2$$ Expand the squares and you are done.

I am sure that you can take from here.