How to teach quadratics: expand, factorise and solve

Year 9 (ages 14 to 15)

Quick answer

A quadratic expression has an x^2 term. Expanding two brackets into one quadratic, factorising a quadratic back into two brackets, and solving a quadratic equation by factorising are three connected skills, each the reverse of the last.

Teach the whole lesson from our teaching unitA textbook-grade, teach-from-this unit: real-world hook, diagrams, worked examples, misconceptions, guided practice and an exit ticket.

How to teach it

Teach expanding first: (x + a)(x + b) = x^2 + (a+b)x + ab, every term in one bracket times every term in the other.
Show factorising as the reverse: for x^2 + Bx + C, find two numbers that ADD to B and MULTIPLY to C.
Introduce solving only once factorising is secure: factorise, then set each bracket to zero separately (if AB = 0, then A = 0 or B = 0).
Always check a factorisation by re-expanding it, and check a solution by substituting it back in.
Keep to monic quadratics (coefficient of x^2 is 1) with whole-number roots until the pattern is fully secure.

Worked example

Factorise x^2 + 2x - 8
Need two numbers that multiply to -8 and add to 2: 4 and -2
So x^2 + 2x - 8 = (x + 4)(x - 2)
Solve = 0: x + 4 = 0 gives x = -4; x - 2 = 0 gives x = 2

Common mistakes

Mixing up which operation goes with which term: the two numbers ADD to the middle term, MULTIPLY to the constant.
Reading (x + 3)(x + 4) = 0 as 'x = 3 and x = 4 at the same time' instead of 'x = -3 or x = -4'.
Assuming every quadratic factorises into whole numbers, when many do not.
Forgetting to check a factorisation by re-expanding it.

Frequently asked questions

What is the pattern for expanding (x + a)(x + b)?

It always expands to x^2 + (a+b)x + ab: the x^2 term comes from x times x, the middle term's coefficient is the SUM of a and b, and the constant term is their PRODUCT.

How do you factorise a quadratic like x^2 + 7x + 12?

Find two numbers that multiply to the constant term (12) and add to the middle coefficient (7). Here that is 3 and 4, so it factorises as (x + 3)(x + 4), the reverse of expanding.

How do you solve a quadratic equation by factorising?

Factorise the quadratic into two brackets, then use the fact that if two things multiply to zero, at least one of them must be zero. (x - 3)(x + 2) = 0 means x = 3 or x = -2.

What year is expanding, factorising and solving quadratics taught?

In the Australian Curriculum this is a Year 9 skill: AC9M9A02 covers expanding and factorising, and AC9M9A04 covers graphing and solving quadratic equations, for monic quadratics with integer roots.

Practise with free worksheets

Printable worksheets with answer keys that are never wrong.

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