How to read sentences with three quantifiers and check for their truth?

I am trying to solve an exercise in which I must decide whether the following sentences are true:

• $$\forall b \; \exists a \; \forall x \; (x^2 + ax + b > 0)$$

• $$\exists b \; \forall a \; \exists x \; (x^2 + ax + b = 0)$$

• $$\exists a \; \forall b \; \exists x \; (x^2 + ax + b = 0)$$

But I don't know how to parse them to check if they are indeed true. I guess I know how to parse two quantifiers easily but this is kinda confusing to me.

• As for how to read them, try reading the qualifiers aloud in order. For example, the first one is of the form "For every person $b$, there exists some city $a$ such that for every shop $x$, they have bought something there". Specifically, the first statement says that if you pick any old $b$, you will be able to then find some $a$ (possibly depending on $b$) that makes the statement $x^2 + ax + b > 0$ true for all $x$. Can informally judge whether or not that statement is true? Jan 26 '20 at 23:00

For the first one, it reads as follows:

For any $$b$$, there exist $$a$$ such that for any $$x$$, $$x^2 + ax + b > 0.$$

Perhaps by graphing this you can see this statement is most certainly false for $$b \le 0$$. However, to show the statement is false, we can show that its negation is true.

Negation: There exists $$b$$ such that for any $$a$$, there exist an $$x$$ such that $$x^2 + ax + b \le 0.$$

So, it suffices to show that the negation is true. Take $$b=0$$ and for any $$a$$ choose $$x=0$$. Then $$x^2 + ax = x(x+a) = 0 \le 0$$.

Hopefully you can see now how to do the other two.

I often find it helpful to mentally insert "such that" after $$\exists$$.

The first example you gave I would read as "for all $$a$$, there exists a $$b$$ such that for all $$x$$ ...."

Another way to think you this, is that I am giving you an $$a$$, and you have to find a $$b$$ such that if I then give you any $$x$$, the rest of the statement will hold.

Example 2 would read "there exists a $$b$$ such that for every $$a$$, there exists an $$x$$ such that..."

And example three would be the same, but with $$a$$ and $$b$$ swapped around.

Let me know if this helps!