This Hint is for students who have only recently met complex numbers.

What can you say about the product of the roots of the quadratic equation $x^2 + px + q = 0$?

If the roots of this quadratic equation, where $p$ and $q$ are real coefficients, are the complex numbers $z_1$ and $z_2$ explain why $z_1$ and $z_2$ are complex conjugates.

Explain why the square of the distance from the origin of the point $u+iv$ in the Argand diagram is given by the product of complex conjugates $(u+iv)(u-iv)$.

What does this tell you about the locus of the complex roots of $x^2 +px +q = 0$ as you change $p$ keeping $q$ constant?

Was your conjecture correct?