Factorizing Quadratic Expressions by Completing the Square: MYP 4 Mathematics Notes

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.

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.

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

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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