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?