Why do this problem?

Here you have an interactivity which shows how the roots of a quadratic equation change continuously as the coefficients in the equation (and hence the graph) change. Learners can investigate for themselves just when the roots are real, when they are coincident and when they are complex.

Learners can discover the discrimant for themselves from the track and shading in the diagram showing the coefficients.

Learners can discover complex numbers for themselves. The complex roots of the quadratic equation appear in the Argand diagram and as you change the quadratic so you see the complex roots change.

Possible approach

This activity can be used to encourage independent learning as the text guides (scaffolds) learners through a discovery process. The teacher might ask the learners to work independently or in pairs and stop the class at certain points so that they can share their findings and so that the teacher can emphasise the important ideas.

If some of the class have learnt the formula for the roots of the quadratic equation and the significance of the discriminant, then this activity will re-force and extend what they know. If they have only met the form $ax^2 + bx + c =0$ then the fact that this can be reduced to two coefficients needs to be pointed out and the link made to the form $x^2 + px + q = 0$.

Key questions

What happens to the graph of $y=x^2 + px + q$ as you change
the coefficients?

What do you notice about the intersections of the graph of
$y=x^2 + px + q$ with the $x$ axis.

Identify two different regions in the $(p,q)$ plane shown the
coloured track left by the point $(p, q)$ as you move it around.
Can you explain the significance of the two regions and the
boundary between them?

The complex roots of the quadratic equation appear in the
green Argand diagram. What happens to the roots as you change the
quadratic?