Quadratic Equations and their Roots

Question

I have a maths competition coming up soon and they've given me some practice problems but I seem to be stuck half way on them. Maybe I just don't understand the question clearly. I was really hoping someone could help me clearly understand this question. Suppose r and s are the solutions for the quadratic equation x^2+Ax+B and the equation x^2+Cx+D=0 has repeated root r-s. Express D in terms of A and B.

Answer 1

By Vieta's Formula's

$$r+s=-A$$ $$rs=B$$

Since $$(x-y)^2=(x+y)^2-4xy$$

As $$D=(r-s)^2=(r+s)^2-4rs=A^2-4B$$

Answer 2

Express what is given:

$$x^2+Ax+B=(x-r)(x-s)=x^2-(r+s)x+rs$$ and $$x^2+Cx+D=(x-(r-s))^2=x^2-2(r-s)x+(r-s)^2.$$

Then for $D$ in terms of $A$ and $B$, what is the relation between $(r-s)^2$ and $-(r+s), rs$ ?