Factorising Harder Quadratics

Factorising Harder Quadraticsvideo

Factorising harder quadratics

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 $4x^2-25x-21$

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

  • 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

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

    • $4x^2-28x+3x-21$

  • 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

    • $4x\left(x-7\right)+3\left(x-7\right)$

  • These terms now have a common factor of $\left(x-7\right)$

    • This whole bracket can be factorised out

    • This gives the answer $\left(x-7\right)\left(4x+3\right)$

Method 2: Factorising using a grid

• Use the same example: factorising $4x^2-25x-21$

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

  • 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

  • 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:

  • $\left(x-7\right)\left(4x+3\right)$