What is Completing the Square?

When we work with quadratic expressions, we often want to rewrite them in a more useful form. Completing the square is a powerful algebraic technique that rewrites a quadratic expression of the form $a x^{2} + b x + c$ into the form $a (x + p)^{2} + q$ .

Completing the Square

A method of rewriting a quadratic expression $a x^{2} + b x + c$ into the equivalent form $a (x + p)^{2} + q$ , where $p$ and $q$ are constants. This reveals the structure of the quadratic and makes it much easier to solve equations, sketch graphs, and identify key features.

Think of it as a transformation — you're not changing the value of the expression, just writing it in a different, more revealing way.

Analogy

Imagine rearranging the same set of furniture in a room. The furniture hasn't changed, but the new arrangement makes the room more functional and easier to understand. Completing the square does exactly this with algebraic expressions — same expression, better arrangement.

This technique is especially useful for:

Solving quadratic equations when factorisation into simple brackets doesn't work neatly
Finding the vertex of a parabola when sketching graphs
Deriving the quadratic formula (which is itself obtained by completing the square!)
Analysing the minimum or maximum value of a quadratic function

Warning

A note on terminology: Despite the subtopic title mentioning "factorising", completing the square is not the same as factorising. Factorising produces a product of linear factors, e.g. $(x + 3) (x - 1)$ . Completing the square produces vertex form $a (x + p)^{2} + q$ — a different, equally useful structure. You can use the completed square form as a step toward finding roots, but the technique itself is about rewriting, not factorising.

Reviewing Perfect Square Trinomials

Before learning the technique, it helps to recognise the pattern at the heart of completing the square: the perfect square trinomial.

When you expand a squared bracket, you get a recognisable pattern:

$(x + p)^{2} = x^{2} + 2 p x + p^{2}$

$(x - p)^{2} = x^{2} - 2 p x + p^{2}$

Notice the key relationships:

The coefficient of $x$ is $2 p$
The constant term is $p^{2}$
So the constant term is always $(\frac{coefficient of x}{2})^{2}$

Example

Recognising perfect square trinomials:

$x^{2} + 6 x + 9 = (x + 3)^{2}$ ✓ because $(\frac{6}{2})^{2} = 9$

$x^{2} - 10 x + 25 = (x - 5)^{2}$ ✓ because $(\frac{10}{2})^{2} = 25$

$x^{2} + 4 x + 4 = (x + 2)^{2}$ ✓ because $(\frac{4}{2})^{2} = 4$

But $x^{2} + 6 x + 7$ is not a perfect square trinomial because $7 \neq = 9$ . This is where completing the square comes in — we adjust it to create the perfect square!

Exam Tip

Memorise this: the constant in a perfect square trinomial is always the square of half the $x$ -coefficient. This is the key insight behind the entire technique.

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