For A:
If , then .
Two identical roots: .
For B:
I don't know precisely what do you mean in "leaving the function unchanged", but I guess you are looking for a condition that makes f(x)=f(-x). So what do we need for ?
Suppose for the moment that b^2-4ac >0, So that f(x) has two distinct real roots , and a>0.
A) Consider the transformation f(x)----> f(x)+k. What is the value of K such that f(x)+k has two identical real roots: Hint, you will need to complete squares.
B) What is the condition on f(x)=ax^2+bx+c such that f(x) -----> f(-x) leaves the function unchanged? Hint: draw a picture of a quadratic function that does not change under this transformation. What kind of transformation is this?