Factorising Quadratics | Mathematics for GCSE/IGCSE - Year 11 PDF Download

Factorising Simple Quadratics

What is a quadratic expression?

• A quadratic expression takes the form: ax² + bx + c (where a ≠ 0). 
• It ceases to be quadratic if it involves higher powers of x, such as x³.
• Monic vs. Non-Monic Quadratics:
  • In the case where a = 1, it's termed a "monic" quadratic expression.
  • For a ≠ 1, it's termed a "non-monic" quadratic expression.

How to Factorize Quadratics

Method 1: Factorizing "by inspection"

• This method involves finding a pair of numbers that multiply to a constant 'c' and add up to 'b'.
• For example, consider the quadratic expression: x² - 2x - 8.
• To factorize, we need to find two numbers that multiply to -8 and add up to -2.
• These numbers are -4 and 2, which satisfy the conditions.
• Write these numbers in a pair of brackets, like this: (x - 4)(x + 2).

Method 2: Factorising "by grouping"

• This is best demonstrated with an example: factorizing x² − 2x − 8.
• We need a pair of numbers that, for x² + bx + c,
  • multiply to c (in this case, -8)
  • add to b (in this case, -2)
• The numbers 2 and -4 meet these conditions.
• Rewrite the middle term using 2x and −4x: x² + 2x − 4x − 8
• Group and factorize the first two terms using 𝑥x as the common factor, and the second two terms using −4 as the common factor: x(x + 2) − 4(x + 2)
• Notice that both groups have a common factor of (x + 2), which can be factored out: (x + 2)(x − 4)

Method 3: Factorizing by Using a Grid

• This is best demonstrated with an example: factorizing x² − 2x − 8.
• We need a pair of numbers that, for x² + bx + c,
  • multiply to c (which is -8)
  • add to b (which is -2)
• The numbers -4 and +2 meet these conditions.
• Write the quadratic equation in a grid, splitting the middle term as −4x and 2x:
  
• Write a heading for the first row, using 𝑥x as the highest common factor of x² and −4x:
  
• Use this to find the headings for the columns, by asking "What does 𝑥x need to be multiplied by to give x²?":
  
• Fill in the remaining row heading using the same idea, asking "What does 𝑥x need to be multiplied by to give 2x?":
  
• 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 the quickest and is recommended for simple quadratics (where a = 1) in an exam.
• However, the other two methods (grouping or using a grid) can be used for more difficult quadratic equations where a ≠ 1, so you should learn at least one of them as well.

Factorising Harder Quadratics

How to Factorise a Harder Quadratic Expression?

Factorising a ≠ 1 "by grouping"

• This is best demonstrated with an example: factorizing 4x² − 25x − 21.
• We need a pair of numbers that, for ax² + bx + c,
  • multiply to ac (which is 4× − 21 = −84)
  • add to b (which is -25)
• The numbers -28 and +3 meet these conditions.
• Rewrite the middle term using −28x and +3x: 
  4x² − 28x + 3x − 21
• Group and factorize the first two terms using 4x as the highest common factor, and the second two terms using 3 as the factor: 
  4x(x − 7) + 3(x − 7)
• Notice that these terms now have a common factor of (x − 7), so this whole bracket can be factored out, leaving 4x + 3 in its own bracket: 
  (x − 7)(4x + 3)

Factorising a ≠ 1 "by using a grid"

• This is best demonstrated with an example: factorizing 4x² − 25x − 21.
• We need a pair of numbers that, for ax² + bx + c,
  • multiply to 𝑎𝑐ac (which in this case is 4× − 21 = −84)
  • add to b (which in this case is -25)
• The numbers -28 and +3 meet these conditions.Write the quadratic equation in a grid, splitting the middle term as −28x and +3x:
  
• Write a heading for the first row, using 4x as the highest common factor of 4x² and −28x:
  
• Use this to find the headings for the columns, by asking "What does 4x need to be multiplied by to give 4x²?":
  
• Fill in the remaining row heading using the same idea, asking "What does x need to be multiplied by to give +3x?":
  
• Now, read off the factors from the column and row headings: (x − 7)(4x + 3)

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
  • Since ab is the same quantity as ba, −ab and +ba cancel out:
  • = a² - b²
• From the working above, the difference of two squares, a² − b², factorizes to:
  (a + b)(a − b)
• 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.

Quadratics Factorising Methods

How can we determine if a quadratic expression factorises? 

• Method 1: Utilize a calculator to solve the quadratic expression set to 0. If the solutions are integers or fractions (without square roots), then the quadratic expression can be factorised.
• Method 2: Determine the value under the square root in the quadratic formula, b² - 4ac (known as the discriminant). If this value is a perfect square number, then the quadratic expression factorises.

What factorization methods can be applied to quadratic expressions?

• Does it have 2 terms only?
  • Yes, like x² − 7x:
    • Use "basic factorization" to take out the highest common factor:
      x(x − 7)
  • Yes, like x² − 9:
    • Use the "difference of two squares" to factorize:
      (x + 3)(x − 3)
• Does it have 3 terms?
  • Yes, starting with x², like x² − 3x − 10
    • Use "factorizing simple quadratics" by finding two numbers that add to -3 and multiply to -10: 
      (x + 2)(x − 5)
  • Yes, starting with ax², like 3x² + 15x + 18
    • Check if the 3 in front of x² is a common factor for all three terms (which it is in this case), then use "basic factorization" to factor it out first: 
      3(x² + 5x + 6)
    • The quadratic expression inside the brackets is now x²+…, which factorizes more easily:
      3(x + 2)(x + 3)
  • Yes, starting with ax², like 3x² − 5x − 2
    • The 3 in front of x² is not a common factor for all three terms:
      • Use "factorizing harder quadratics", such as factorizing by grouping or using a grid: (3x + 1)(x − 2)