Quadratic Equations · Grade 9

How do you factorise an algebraic expression?

Factorising is expanding brackets in reverse: it turns a sum into a product. Always try in the same order — common factor first, then a special product formula, and only if neither fits, factorise the trinomial by finding two numbers whose product is the constant and whose sum is the middle coefficient.

Learning objectives

What factorising actually is

Expanding turns a product into a sum. Factorising does the opposite: it returns a sum to the form of a product. The same expression, written two ways.

That form is worth so much because of one property of zero: a product equals zero only if one of its factors is zero. This is why this lesson comes before solving equations rather than after.

The order to try

Common factor first. If every term shares a divisor — a number or a variable — take it outside. This shortens what remains and sometimes finishes the job.

Formula second. One look tells you whether it is a difference of squares or a perfect square, and both factorise immediately with no searching.

Only then the trinomial. Look for two numbers whose product is the constant term and whose sum is the coefficient of x, and write two brackets containing them.

How to find the two numbers

Start from the product, not the sum, because a number has only a few pairs of divisors and infinitely many pairs that add to it. List the divisor pairs and check which gives the right sum.

The signs tell you a lot. A positive product means the two numbers share a sign; a negative product means their signs differ. That halves the search before it starts.

Worked examples

  1. Factorise 3x squared plus 12x

    1. Check for a common factor: both terms have a 3 and an x
    2. Take 3x outside the brackets
    3. Inside remains x plus 4

    Answer: 3x times the bracket (x + 4)

  2. Factorise x squared minus 25

    1. There is no common factor
    2. This is a difference between two squares: x squared and 5 squared
    3. Factorise into a sum times a difference

    Answer: (x + 5)(x − 5)

  3. Factorise x squared plus 7x plus 12

    1. No common factor and no formula fits, so factorise the trinomial
    2. Look for two numbers with product 12 and sum 7
    3. Divisor pairs of 12: 1 and 12, 2 and 6, 3 and 4
    4. The pair 3 and 4 adds to 7

    Answer: (x + 3)(x + 4)

Common mistakes

Skipping the common factor
Without it the trinomial stays large and the numbers are hard to find. Taking a common factor out first almost always turns the rest into an easy exercise, which is why it comes first in the order.
Starting from the sum instead of the product
A given sum has infinitely many pairs; a product has very few. The list of divisor pairs is always short, and checking against it is quick.
Ignoring the signs when choosing the pair
A negative product requires opposite signs and a positive product requires matching ones. Ignoring that leads to brackets that look right and expand into a different expression.

What to remember

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