A4e – Factorising quadratics

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Factorising quadratics

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  • Fully factorise \(x^4-1\)
Hover for hint:
  •   Define \(u = x^2\). Then \(u^2=x^4\).
Hover for answer:
  •   \(x^4-1 \equiv u^2-1 \equiv (u+1)(u-1) \equiv (x^2+1)(x^2-1) \equiv (x^2+1)(x+1)(x-1)\)

  1. Expand \((x+\frac{1}{2})(x+6)\). Is it possible factorise this expanded expression so that the factors only contain integer coefficients and constants?
  2. Expand \((x+\frac{1}{2})(2x+6)\). Is it possible factorise this expanded expression so that the factors only contain integer coefficients and constants?
Hover for answers:
  1.   \((x+\frac{1}{2})(x+6)\equiv x^2+\frac{13}{2}x+3\); no
  2.   \((x+\frac{1}{2})(2x+6)\equiv 2x^2+7x+3 \equiv (2x+1)(x+3)\)

Stan and Ollie are trying to factorise \(x^2-5x+6\). Stan writes \((x-2)(x-3)\). Ollie writes \((2-x)(3-x)\). Are either of them correct? If so, who?

Hover for answers:

  •   They’re both correct, but Stan’s factorisation is more conventional.
Teacher resources

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  • Slides in PDF (one slide per page, suitable for importing into IWB software)
Links to past exam and UKMT questions

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Unlimited practice questions
In the real world
In lesson A18a, you will see how factorising quadratic expressions will help you solve quadratic equations. The ‘In the real world‘ section for that lesson shows why quadratics are so relevant in the real world.