Real(ly) Numbers

If x, y and z are real numbers such that: x + y + z = 5 and xy + yz + zx = 3. What is the largest value that any of the numbers can have?

Roots and Coefficients

If xyz = 1 and x+y+z =1/x + 1/y + 1/z show that at least one of these numbers must be 1. Now for the complexity! When are the other numbers real and when are they complex?

Pair Squares

The sum of any two of the numbers 2, 34 and 47 is a perfect square. Choose three square numbers and find sets of three integers with this property. Generalise to four integers.

Stage: 5 Challenge Level:

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 $x^2 + px + q = (x - \alpha)(x - \beta) = 0$?

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?

Possible extension
The next step is to explore the relationship between the graph of a quadratic function and the roots of the corresponding quadratic equation. The interactive Root Tracker allows you to move the graph around and see how the coefficients and the roots change as the graph changes. Moreover Root Tracker introduces the idea of complex numbers as solutions of quadratic equatons.

Possible support
Power Quady is another non-standard problem on quadtatic equations which calls for some logical reasoning to account for all possible cases and gives practice in algebraic manipulation and in solving quadratic equations.