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Quadratic Denominators (CIE A Level Maths: Pure 3)
Revision Note
Quadratic Denominators
What is meant by partial fractions with quadratic denominators?
- For linear denominators the denominator of the original fraction can be factorised such that the denominator becomes a product of linear terms of the form
- With squared linear denominators, the same applies, except that some (usually just one) of the factors on the denominator may be squared, i.e.
- In both the above cases it can be shown that the numerators of each of the partial fractions will be a constant
- For this course, quadratic denominators refer to fractions that have one linear factor and one quadratic factor (that cannot be factorised) on the denominator
- the denominator of the quadratic partial fraction will be of the form ; very often leaving it as
- the numerator of the quadratic partial fraction could be of linear form,
How do I find partial fractions involving quadratic denominators?
- STEP 1 Factorise the denominator as far as possible (if not already done so)
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- Sometimes the numerator can be factorised too
- STEP 2 Split the fraction into a sum with
- the linear denominator having an (unknown) constant numerator
- the quadratic denominator having an (unknown) linear numerator
- STEP 3 Multiply through by the denominator to eliminate fractions
- STEP 4 Substitute values into the identity and solve for the unknown constants
- Use the root of the linear factor as a value of to find one of the unknowns
- Use to find another one of the unknowns
- Use any value of (keep it small and simple) to find the final unknown
- If the linear factor is then you'll need to use any two other values of to form simultaneous equations
- STEP 5 Write the original as partial fractions
- In harder problems there may be more than one linear or quadratic factor
- In such cases, values of , whatever order they’re used in, will not always eliminate all but one of the unknowns
- Simultaneous equations will need to be used
Worked example
Examiner Tip
- You can check your final answer by substituting a value of x in to both the left and right-hand sides and seeing if they’re equal
- Choose a small value of x to keep things simple but not a value that would make a denominator zero
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