Factorising Quadratics (AQA GCSE Further Maths)

Factorising Simple Quadratics

What is a quadratic expression?

• A quadratic expression is in the form:

  • ax² + bx + c (as long as a ≠ 0)

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

• If a = 1 e.g. , it can be called a “monic” quadratic expression

• If a ≠ 1 e.g. , it can be called a “non-monic” quadratic expression

Method 1: Factorising "by inspection"

• This is shown easiest through an example; factorising 

• We need a pair of numbers that for 

  • multiply to c

    • which in this case is -8

  • and add to b

    • which in this case is -2

  • -4 and +2 satisfy these conditions

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

    • 

Method 2: Factorising "by grouping"

• This is shown easiest through an example; factorising 

• We need a pair of numbers that for 

  • multiply to c

    • which in this case is -8

  • and add to b

    • which in this case is -2

  • 2 and -4 satisfy these conditions

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

    • 

  • Group and factorise the first two terms, using x as the highest common factor, and group and factorise the second two terms, using -4 as the factor

    • 

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

    • 

Method 3: Factorising "by using a grid"

• This is shown easiest through an example; factorising 

• We need a pair of numbers that for 

  • multiply to c

    • which in this case is -8

  • and add to b

    • which in this case is -2

  • -4 and +2 satisfy these conditions

  • Write the quadratic equation in a grid (as if you had used a grid to expand the brackets), splitting the middle term as -4x and 2x

  • 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?”

• 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?”

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

• However the other two methods (grouping, or using a grid) can be used for harder quadratic equations where a ≠ 1 so you should learn at least one of them too

Factorising Harder Quadratics

How do I factorise a harder quadratic expression?

Factorising a ≠ 1 "by grouping"

• This is shown easiest through an example; factorising 

• We need a pair of numbers that for 

  • multiply to ac

    • which in this case is 4 × -21 = -84

  • and add to b

    • which in this case is -25

  • -28 and +3 satisfy these conditions

  • Rewrite the middle term using -28x and +3x

    • 

  • Group and factorise the first two terms, using 4x as the highest common factor, and group and factorise the second two terms, using 3 as the factor

    • 

  • Note that these terms now have a common factor of (x - 7) so this whole bracket can be factorised out, leaving 4x + 3 in its own bracket

    • 

Factorising a ≠ 1 "by using a grid"

• This is shown easiest through an example; factorising 

• We need a pair of numbers that for 

  • multiply to ac

    • which in this case is 4 × -21 = -84

  • and add to b

    • which in this case is -25

  • -28 and +3 satisfy these conditions

  • Write the quadratic equation in a grid (as if you had used a grid to expand the brackets), splitting the middle term as -28x and +3x

  • 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 4x as the highest common factor of 4x² and -28x

• You can then use this to find the headings for the columns, e.g. “What does 4x need to be multiplied by to give 4x²?”

• 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 +3x?”

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

  • 

How do I factorise a quadratic with two variables?

• To factorise 3x² + 13xy - 10y²

  • Factorise the easier quadratic 3x² + 13x - 10

    • (3x - 2)(x + 5)

  • Insert y's on the last terms in the brackets

    • (3x - 2y)(x + 5y)

• Check by expanding (3x - 2y)(x + 5y)

  • 3x² + 15xy - 2yx - 10y²

    • 3x² + 13xy - 10y²  ✓

Difference Of Two Squares

What is the difference of two squares?

• When a "squared" quantity is subtracted from another "squared" quantity, you get the difference of two squares

  • for example,

    • a² - b²

    • 9² - 5²

    • (x + 1)² - (x - 4)²

    • 4m² - 25n², which is (2m)² - (5n)²

How do I factorise the difference of two squares?

• Expand the brackets (a + b)(a - b)

  • = a² - ab + ba - b²

  • ab is the same quantity as ba, so -ab and +ba cancel out

  • = a² - b²

• From the working above, the difference of two squares, a² - b², factorises to

• It is fine to write the second bracket first, (a - b)(a + b)

  • but the a and the b cannot swap positions

    • a² - b² must have the a's first in the brackets and the b's second in the brackets

• It might not be obvious that you can use the difference of two squares

  • Try factoring out any common factors first

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