Why do this problem?
The activity involves using the sum and product of the roots of a
quadratic equation, the relationship between the coefficients and
the roots, the discriminant of a quadratic equation, solving
simple quadratic equations and checking the solutions by
substitution in the original equation. It gives practice and
re-inforcement of these ideas without being tedious.
The non-standard nature of the problem encourages learners to
think for themselves so developing reasoning and problem solving
skills which are transferable to other situations. It is also
necessary to work systematically through all possible cases to
find all solutions.
Possible approach
Suggest the learners try to find some solutions for themselves,
initially by trial and error, and to check their solutions by
substitution in the original equations. Then ask how they might
make sure they have found all the solutions.
It is sometimes useful to list what you know and then to try to
apply that knowledge. In this case, as each of the two equations
have coefficients that occur in the other equation, and the same
roots, solving the problem must involve the relationship between
the cofficients and the sum and products of the roots.
This problem can be used to motivate the neeed to know and use
this relationship or as re-inforcement of the ideas or as a
revision exercise.
Key questions
What are the roots of the quadratic equation
?
What is the relationship between the sum of the roots and the cofficients
in the equation?
What is the relationship between the productof the roots and the cofficients
in the equation?