Evaluating Expressions Given in Terms of Roots of Quadratic Equations

If a quadratic equationhas rootsandthen we can write the equation asExpanding the brackets gives

(1)

Suppose that we have the equationwith rootsandthen comparing with (1) givesand

We can use simple algebra to evaluate many expressions given in terms ofandwithout ever knowing what the values ofand

Example:

Example:

Example:

Example:

Example:

Any function symmetric in terms ofandcan be evaluated in this way. Symmetric here means that ifandare interchanged then the expression is unchanged:

This is the third example above.