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Roots of Quadratic Equations
We recall that for 
whose roots are 
and 
, sum of roots 
and product of roots 
.
It is a common difficulty for students to form quadratic equations whose roots are more complicated, e.g.
, 
.
The strategy is to find the sum and product of the new roots and make use of the original values of
, 
and a very useful identity ^2 - 2\alpha\beta)
.
Let's apply the above method for our example: The equation
has roots and . Form an equation whose roots are and .
Step 1: Find and based on the original equation.
 = 2)
 + \left(\beta + \frac{1}{\beta}\right) = \left(\alpha + \beta\right) + \left(\frac{1}{\alpha} + \frac{1}{\beta} \right))
 + \left(\frac{\beta + \alpha}{\alpha\beta} \right))
\left(\beta + \frac{1}{\beta} \right) = \alpha\beta + \frac{\alpha}{\beta} + \frac{\beta}{\alpha} + \frac{1}{\alpha\beta})
^2 - 2\alpha\beta}{\alpha\beta} + \frac{1}{\alpha\beta})
It is a common difficulty for students to form quadratic equations whose roots are more complicated, e.g.
The strategy is to find the sum and product of the new roots and make use of the original values of
Let's apply the above method for our example: The equation
Step 1: Find
Step 2: Find sum and product of new roots.
Step 3: Form the new equation.
The required equation is 
.
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, Nov 17 2009, 7:00 AM EST
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