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.
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
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 $x^2 + px + q = 0$.
What happens to the graph of $y=x^2 + px + q$ as you change
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