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
ax2 + 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 x2 + px + q = 0.
Key questions
What happens to the graph of y=x2 + px + q as you change the coefficients?
What do you notice about the intersections of the graph of
y=x2 + 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?