Factorising quadratics ax²+bx+c (SQA National 5 Maths): Revision Note

Factorising quadratics ax²+bx+c

How do I factorise a quadratic expression where a ≠ 1 in ax² + bx + c?

Method 1: Factorising by grouping

• This is shown most easily through an example: factorising 

• We need a pair of numbers that, for 

  • both multiply to give ac

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

  • and both add to give b

    • b in this case is -25

  • -28 and +3 satisfy these conditions

    • (-28) × 3 = -84

    • (-28) + 3 = -25

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

    • 

  • Group and fully factorise the first two terms, using 4x as the common factor

  • and group and fully factorise the last two terms, using 3 as the common factor

    • 

  • These terms now have a common factor of

    • This whole bracket can be factorised out

    • This gives the answer

Method 2: Factorising using a grid

• Use the same example: factorising 

• We need a pair of numbers that for 

  • multiply to give ac

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

  • and add to give b

    • b in this case is -25

  • -28 and +3 satisfy these conditions

    • (-28) × 3 = -84

    • (-28) + 3 = -25

  • Write the quadratic expression in a grid

    • (as if you had used a grid to expand the brackets)

    • splitting the middle term up as -28x and +3x (either order)

  • 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 brackets from the column and row headings:

  •