Factorising Simple Quadratics (Cambridge International IGCSE Maths)

Factorising simple quadratics

What is a quadratic expression?

• A quadratic expression is in the form:

  • ax² + bx + c (where a ≠ 0)

• 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 $x^{2}-2x-8$

• We need a pair of numbers that for $x^{2}+bx+c$

  • 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:

    • (x+2)(x−4)

How do I factorise quadratics by grouping?

• This is shown most easily through an example: factorising $x^{2}-2x-8$

• We need a pair of numbers that for $x^{2}+bx+c$

  • 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

    • $x^{2}+2x-4x-8$

  • 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

    • x (x+2)−4 (x+2)

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

    • (x+2)(x−4)

How do I factorise quadratics using a grid?

• This is shown most easily through an example: factorising $x^{2}-2x-8$

• We need a pair of numbers that for $x^{2}+bx+c$

  • 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

  • (x+2)(x−4)

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

Exam Hint As a check, expand your answer and make sure you get the same expression as the one you were trying to factorise.

Worked Example (a) Factorise $x^{2}-4x-21$.

Answer:

We will factorise by inspection

We need two numbers that multiply to give -21, and sum to give -4 +3 and -7 satisfy this

$3 \times ( - 7 ) = - 21$

$> 3 + ( - 7 ) = - 4$

Write down the brackets

Final answer: (x + 3)(x - 7)

(b) Factorise $x^{2}-5x+6$.

Answer:

We will factorise by splitting the middle term and grouping

We need two numbers that multiply to 6, and sum to -5 -3 and -2 satisfy this

$( - 3 ) \times ( - 2 ) = 6$

$> ( - 3 ) + ( - 2 ) = - 5$

Split the middle term

x² - 2x - 3x + 6*

Factorise x out of the first two terms

x(x - 2) - 3x +6*

Factorise -3 out of the last two terms

x(x - 2) - 3(x - 2)*

These have a common factor of (x - 2) which can be factored out

Final answer: (x - 2)(x - 3)

(c) Factorise $x^{2}-2x-24$.

Answer:

We will factorise by using a grid

We need two numbers that multiply to -24, and sum to -2 +4, and -6 satisfy this

$4 \times ( - 6 ) = - 24$

$> 4 + ( - 6 ) = - 2$

Use these to split the -2x term and write in a grid

	x2 *	* +4x
	*-6x	-24

Write a heading using a common factor for the first row

x	x2 *	* +4x
* *	*-6x	-24

Work out the headings for the rows “What does x need to be multiplied by to make x²?” “What does x need to be multiplied by to make +4**x?”

* *	x	+4
x	x2 *	* +4x
* *	*-6x	-24

Repeat for the heading for the remaining row “What does x need to be multiplied by to make -6x?” (Or “What does +4 need to be multiplied by to make -24?”)

* *	x	+4
x	x2 *	* +4x
-6	*-6x	-24

Read off the factors from the column and row headings

Final answer: (x + 4)(x - 6)