1.3.3 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)

• • 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

Exam Tip