Factorising Quadratics (Cambridge O Level Maths)

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

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
    • 

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

How do I know if it factorises?

• Method 1: Use a calculator to solve the quadratic expression equal to 0
  • If the solutions are integers or fractions (without square roots), then the quadratic expression factorises
• Method 2: Find the value under the square root in the quadratic formula, b² – 4ac (called the discriminant)
  • If this number is a perfect square number, then the quadratic expression factorises

Which factorisation method should I use for a quadratic expression?

• Does it have 2 terms only?
  • Yes, like 
    • Use "basic factorisation" to take out the highest common factor
    • 
  • Yes, like 
    • Use the "difference of two squares" to factorise
    • 
• Does it have 3 terms?
  • Yes, starting with x² like
    • Use "factorising simple quadratics" by finding two numbers that add to -3 and multiply to -10
    • 
  • Yes, starting with ax² like
    • Check to see if the 3 in front of x² is a common factor for all three terms (which it is in this case), then use "basic factorisation" to factorise it out first
    • 
    • The quadratic expression inside the brackets is now x² +... , which factorises more easily
    • 
  • Yes, starting with ax² like
    • The 3 in front of x² is not a common factor for all three term
    • Use "factorising harder quadratics", for example factorising by grouping or factorising using a grid
    •