Hint for (a): a common error is to write this as
$$\forall x\ (\,P(x) \wedge Q(x)\,)\ .$$
However, this means that for all text, it is a clear explanation and it is satisfactory. What you actually want to say is that for all text, if it is a clear explanation then it is satisfactory. Can you do this? (Another hint: you only have to change one symbol in the above incorrect ansswer.)
The statement
There is a student in this class who has taken every course offered by one of the departments in this school
could be "worded" more clearly. I take it to mean that there is some student $x$ and there is some ("one") department $z$, such that $x$ has taken every course $y$ offered by $z$:
$$\exists x\,\exists z\,\forall y\,(O(y,z) \to T(x,y)). \tag{1}
$$
Notice that in the highlighted English sentence, the "every course" quantifier precedes the department quantifier ("one of the departments"), the opposite of their order in the logical structure.
You didn't symbolize "in this class", so I didn't either in (1), but you should. To keep it simple, let $C(x)$" stand for "$x$ is in this class". A better rendering:
$$\exists x\, \big( C(x) \land \exists z\,\forall y\,(O(y,z) \to T(x,y)) \big). \tag{2}
$$
The "domain" over which variables range is a little unusual for a basic logic example but quite typical of "real-world" situations. It's most naturally thought of as multi-sorted: there are students, departments, and courses, so the actual universe of the interpretation is the disjoint union of these sets. In the intended interpretation of $T$, if $T(x,y)[\mathbf{a}/x, \mathbf{b}/y ]$ is true, then $\mathbf{a}$ is a student and $\mathbf{b}$ is a course; similarly, $O(y,z)[\mathbf{b}/y, \mathbf{c}/z ]$ is true only of courses $\mathbf{b}$ and departments $\mathbf{c}$.
Your statement is very different than what the English sentence (tries to) express. Your rendering:
$$\exists x\,\forall y\,\exists z\,(T(x,y) \land O(y,z))
$$
is equivalent to:
$$\exists x\,\forall y\,T(x,y) \land \forall y\,\exists z\,O(y,z)
$$
which says: there's a student who has taken every course, and every course is offered by some department — two independent statements.
Best Answer
For all real $a,b,c,x,y,z$, if
$$(ax^2 + bx + c)^2 + (ay^2 + by + c)^2 + (az^2 + bz + c)^2 = 0,$$ then $x = y$ or $y = z$ or $z = x$.