Factorising quadratics x²+bx+c (SQA National 5 Maths): Revision Note

Factorising quadratics x²+bx+c

What is a quadratic expression?

• A quadratic expression is in the form ax² + bx + c (where a ≠ 0)

  • An expression like this with three terms is sometimes also known as a trinomial

    • If a = 1, then the x² coefficient is said to be unitary

    • If a ≠ 1, then the x² coefficient is said to be non-unitary

• If there are any higher powers of x (like x³ say) then it is not a quadratic

How do I factorise quadratics by inspection?

• This is shown most easily through an example: factorising 

• We need a pair of numbers that for 

  • multiply to give c

    • which in this case is -8

  • and add to give b

    • which in this case is -2

  • +2 and -4 satisfy these conditions

    • 2 × (-4) = -8  and  2 + (-4) = -2

  • Write these numbers in a pair of brackets like this: 

    • 

How do I factorise quadratics by grouping?

• This is shown most easily through an example: factorising 

• We need a pair of numbers that for 

  • multiply to give c

    • which in this case is -8

  • and add to give b

    • which in this case is -2

  • +2 and -4 satisfy these conditions

    • 2 × (-4) = -8  and  2 + (-4) = -2

  • Rewrite the middle term by using +2x and -4x

    • 

  • Group and factorise the first two terms, using x as the common factor

  • and group and factorise the last two terms, using -4 as the common factor

    • 

  • Note that these both now have a common factor of (x + 2) so that whole bracket can be factorised out

    • 

How do I factorise quadratics using a grid?

• This is shown most easily through an example: factorising 

• We need a pair of numbers that for 

  • multiply to give c

    • which in this case is -8

  • and add to give b

    • which in this case is -2

  • +2 and -4 satisfy these conditions

    • 2 × (-4) = -8  and  2 + (-4) = -2

  • Write the quadratic equation in a grid (as if you had used a grid to expand the brackets)

    • splitting the middle term as +2x and -4x

• The grid works by multiplying the row and column headings, to give a product in the boxes in the middle

• Write a heading for the first row, using x as the highest common factor of x² and -4x

• You can then use this to find the headings for the columns

  • e.g. “What does x need to be multiplied by to give x²?”

  • and “What does x need to be multiplied by to give -4x?”

• We can then fill in the remaining row heading using the same idea

  • e.g. “What does x need to be multiplied by to give +2x?”

  • or “What does -4 need to be multiplied by to give -8?”

• We can now read off the factors from the column and row headings

  • 

Which method should I use for factorising simple quadratics?

• The first method, by inspection, is by far the quickest

  • So this is recommended in an exam for simple quadratics (where a = 1)

• However some students find the other methods helpful

  • So you may want to learn at least one of them too

  • The grouping method will also be useful for factorising quadratics where a ≠ 1