Quadratic expressions have a distinct look about them. They have the form:

  • a{x^2} + bx + c \hphantom{aaa} \text{where}\hphantom{aaa}a \ne 0

When it comes to factorising a quadratic expression, there are particular methods depending on the form of the expression

In the tutorials which follow, I will show you how to factorise the above expressions.

HCF types

Your first priority in factorising is to check to see if the quadratic expression has a common factor. In this tutorial I introduce you to what a quadratic expression is and then look at factorising quadratic expressions where there is a highest common factor (HCF).

Examples in the video

Factorise the following:

  1. {x^2} - 3x
  2. 10{a^2} - 15a
  3. 12{x^2} - 6x + 9

Difference of two squares type

Some quadratic expressions do not have a common factor but consist of two terms separated by a minus sign and each term is the square of something. This is known as the difference of 2 squares type.

Examples in the video

Factorise the following:

  1. {x^2} - 9
  2. {y^2} - 1
  3. 16{x^2} - 25{y^2}
  4. {x^4} - 1
  5. 2{x^2} - 72

Trinomials

Some quadratic expressions have 3 terms (trinomials) and no common factor yet can still be factorised.

There are two methods of factorising such expressions. 1) by grouping 2) by inspection. Which method you choose is up to you but I generally prefer method 2 as it can be quicker but requires a bit more practice.

Method 1) – Trinomials Grouping Method

In this video I show you how to factorise the following trinomials by method 1) grouping

Examples in the video

Factorise the following:

  1. 2{x^2} + 13x + 15
  2. 3{x^2} + 5x - 2
  3. 5{x^2} - 14x - 3
  4. 2{x^2} - 7x + 6

(Method 2) – Trinomials Inspection Method

In this video I show you how to factorise the following trinomials by method (2) inspection

Examples in the video

Factorise the following:

  1. {x^2} + 3x + 2
  2. {x^2} - 5x + 6
  3. {x^2} + 2x - 8
  4. {x^2} - 2x - 3

I now extend the work on factorising quadratic trinomials (3 terms) to ones where the first term is not x² only. I would encourage you to look at all the videos as the thinking changes depending on the signs in the trinomial. This is not an easy topic and requires patience.

In the first of these tutorials I show you how to factorise a trinomial with all terms positive.

Examples in the video

Factorise the following:

  1. 2{x^2} + 5x + 3
  2. 3{x^2} + 11x + 6

In the next tutorial I show you how to factorise a trinomial where the 2nd term is negative and remind you of an important stage in the 2nd example.

Examples in the video

Factorise the following:

  1. 5{x^2} - 23x + 12
  2. 10{x^2} - 46x + 24

In the next tutorial I show you how to factorise a trinomial where the 3rd term is negative. An important one to try.

Examples in the video

Factorise the following:

  1. {30{x^2} + 130x - 100}

In the next tutorial I show you how to factorise a trinomial where the last 2 terms are negative.

Examples in the video

Factorise the following:

  1. {7{x^2} - 23x - 20}

In the last tutorial in this section I show you how to factorise a trinomial where the x² term is not a prime number and so can be split into factors. This can often be a lot harder as there are more combinations to try.

Examples in the video

Factorise the following:

  1. {6{x^2} + x - 12}

Sometimes a quadratic trinomial can be disguised as in these two examples below. Check out the tutorial if you cannot factorise them.

Examples in the video

Factorise the following:

  1. {x^4} - 3{x^2} - 10
  2. 2a + \sqrt a  - 21