Factorising Quadratics (DP IB Analysis & Approaches (AA)): Revision Note

Factorising Quadratics

Why is factorising quadratics useful?

• Factorising gives roots (zeroes or solutions) of a quadratic

• It gives the -intercepts of the graph of the quadratic

How do I factorise a monic quadratic of the form x²+bx+c?

• A monic quadratic is a quadratic where the coefficient of the term is 1

• You might be able to spot the factors by inspection

  • Especially if is a prime number

• Otherwise, start by finding two numbers and which have,

  • A sum equal to

    • 

  • A product equal to

    • 

• Rewrite the middle term of the quadratic, , as

• Use this to factorise

• A shortcut is to write down as soon as you have found and

How do I factorise a non-monic quadratic of the form ax²+bx+c?

• A non-monic quadratic is a quadratic where the coefficient of the  term is not equal to 1

• If , and have a common factor then first factorise that out to leave a quadratic with coefficients that have no common factors

  • E.g. can be rewritten as

• You might be able to spot the factors by inspection

  • Especially if and/or are prime numbers

• Otherwise, start by finding two numbers and which have,

  • A sum equal to

    • 

  • A product equal to

    • 

• Rewrite the middle term, , as

• Use this to factorise

• A shortcut is to write

  • 

  • Then factorise common factors from numerator to cancel with the a on the denominator

How do I use the difference of two squares to factorise a quadratic of the form a²x²-c²?

• The difference of two squares can be used when

  • There is no linear term and

  • The constant term is a negative

  • E.g.

• First, square-root the two terms  and 

• The two factors of the quadratic are the sum of the square roots and the difference of the square roots

  • i.e.

  •