How to Complete the Square

Completing the square is a way of rewriting a quadratic expression $ax^2+bx+c$ in the tidy form $a(x+p)^2+q$, often called vertex form. The two forms describe exactly the same curve, but the second one hands you information the first one hides.

From $a(x+p)^2+q$ we can read the turning point of the parabola at once: its coordinates are $(-p,\ q)$. If $a>0$ the curve opens upward and $q$ is the minimum value; if $a<0$ it opens downward and $q$ is the maximum value.

In Add Math this single skill unlocks minimum and maximum problems, the axis of symmetry, sketching parabolas, and even solving quadratic equations when factorising is awkward.

We reach for vertex form because it turns a hard question into a reading exercise. Once a quadratic sits as $a(x+p)^2+q$, the smallest value the squared bracket can take is zero, so the whole expression is never smaller than $q$ when $a>0$, the minimum is handed to you without calculus and without a graph.

This is a core Form 4 skill in the Quadratic Functions chapter, and it feeds directly into the range of a function, the axis of symmetry $x=-p$, and the direction the curve opens. Getting it solid early makes later topics much easier to follow.

Reach for completing the square whenever a question asks for the minimum value, the maximum value, or the turning point of a quadratic function. The wording often says "express in the form $a(x+p)^2+q$" and then "state the minimum point".

It is also the right tool when a quadratic will not factorise neatly and you still need its roots, or when you are asked for the range of a quadratic function. If you see a parabola sketch with its vertex marked, or a phrase like "least value", that is your cue.

Because the vertex form makes the symmetry obvious, quadratic-function questions built on this method appear in both Paper 1 and Paper 2.

Watch also for optimisation-style word problems, greatest area, least cost, maximum height, that hide a quadratic inside a real-world story. Rewriting the expression in vertex form gives the best value directly, and the value of $x$ that achieves it, without any guessing.

Here is the full procedure. When a leading coefficient is present, work in this order:

1. 1

   Make the coefficient of $x^2$ equal to 1

   If $a\neq 1$, factor $a$ out of the $x^2$ and $x$ terms only, and leave the constant outside the bracket.

2. 2

   Halve the coefficient of $x$

   Take the number in front of $x$ inside the bracket, halve it, and call the result $p$. This $p$ is what goes inside $(x+p)^2$.

3. 3

   Add and subtract the square

   Add $p^2$ and subtract $p^2$ inside the bracket so the value is unchanged, then group the first three terms as a perfect square $(x+p)^2$.

4. 4

   Multiply the bracket back out

   Expand the factor $a$ across the $-p^2$ term you created, and combine it with the constant left outside.

5. 5

   Write the vertex form and read the answer

   State the result as $a(x+p)^2+q$; the turning point is $(-p,\ q)$, and $q$ is the minimum (if $a>0$) or the maximum (if $a<0$).

Try this yourself first, then check each line against the solution.

Because the vertex form is now visible, two more answers come for free. The axis of symmetry is the vertical line through the vertex, $x=-2$.

And since the least value of $f$ is $-3$, the range of $f$ is $f(x)\ge -3$. Reading three results from one line of algebra is exactly why examiners like this form.

• Forgetting to factor out $a$ first. When the coefficient of $x^2$ is not 1, you must take it out of the $x^2$ and $x$ terms before halving.
• Halving but not squaring, or squaring but not subtracting. You must add $p^2$ and subtract the same $p^2$ so the expression keeps its value.
• Forgetting to multiply the $-p^2$ by the factor $a$ when it leaves the bracket. This is the most common sign error.
• Reading the vertex as $(p,\ q)$ instead of $(-p,\ q)$. The $x$-coordinate is the value that makes the bracket zero.
• Assuming $q$ is always a minimum. It is only the minimum when $a>0$; when $a<0$ the curve opens downward and $q$ is the maximum.

Almost every lost mark in completing the square comes from one specific line, most often the moment the factor $a$ has to be multiplied back across the $-p^2$ term. In a one-to-one lesson our teacher watches you work that exact line, spots whether the slip is a sign, a missing factor, or a halving error, and has you redo just that step until it is automatic.

Because our teachers are experienced, you get someone who explains the why, not only the how. Lessons are online and taught in English, and because working is marked step by step, we show you how to earn method marks even when the final number goes wrong.

We also build the habit of a ten-second check, expanding the vertex form back to the original, so a careless slip never survives to the final line.