Q:

What is a complex constant and how do I use it?

I have a question I am trying to understand:
"Let $b$ and $c$ be complex constants such that $z^2+bz+c=0$ has two different real roots.  Show that $b$ and $c$ are real."
My biggest problem here is that I don't know what a complex constant is; I have tried to google it, however all I am getting for results is computer programming information, which I am finding unhelpful.  Could someone please explain to me what exactly this is and how I can use it to solve the above problem?

A:

Hint 1: Suppose you have a polynomial with two distinct real roots be labelled as $r_1,r_2$. How would you write this polynomial, and how does this compare with the form given?
Hint 2: Does multiplying or adding real numbers ever produce a complex number?

A:

Using the Hint from Semiclassical with the two real roots $r_1 \neq r_2$:
\begin{align}
r_1^2+r_1(b_1 +ib_2) + (c_1 + ic_2) &= 0\\
r_2^2+r_2(b_1 +ib_2) + (c_1 + ic_2) &= 0
\end{align}
Summing up the imaginary parts
\begin{align}
i(r_1b_2 + c_2) &= 0\\
i(r_2b_2 + c_2) &= 0
\end{align}
So you get either $b_2 = c_2  =0$ or $r_1 = r_2$. Last one would be a contradiction to the statement above. Therefore $b_2 = c_2  =0$ and $b$ and $c$ are real.

