Forming Quadratics with New Roots (Edexcel International A Level Further Maths)
Revision Note
Author
Mark CurtisExpertise
Maths
Forming Quadratics with New Roots
How do I form quadratics with new roots?
The quadratic equation has roots and where
Any new quadratic equations with new roots will have the form
For example, to find the quadratic equation with roots and
The new equation is
You can then work out these coefficients using the identities from before
and
Substitute in and
Which identities do I need to know?
You should know identities involving powers of products of roots
and so on
You should know the identity for the sum of the squares of the roots
which comes from expanding
You should know the identity for the sum of the cubes of the roots
which comes from the binomial expansion of
then rearranging
You should know how to use algebraic fractions (but do not need to learn these identities)
and so on
Exam Tip
Don't forget to add to your quadratic equations!
Read the question carefully to see how to give your final answer
For example, if asked for integer coefficients
Worked Example
The quadratic equation has roots and .
Without solving the equation, find a quadratic equation with roots
and
giving your answer in the form where , and are integers to be determined.
Start with the equation
Use and to find the sum and product of the roots
and
Form the new equation using
Simplify the coefficient of
Add the algebraic fractions
Use (or derive) the identity for the sum of the squares of the roots
Substitute in and
Now simplify the constant term
Expand the brackets and add the algebraic fractions
Use (or derive) the identity for the sum the cubes of the roots
Substitute in and
Write out the new quadratic equation
Simplify the coefficients
The question asks for the equation to have integer coefficients
Multiply both sides by 3
Don't forget to write to get full marks
Any multiple of the answer (e.g. ) would also be accepted
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