Completing the square visualiser
Completing the square is one of those GCSE topics that students can learn as a recipe — halve it, square it, subtract it — without ever understanding why the recipe works. This free interactive tool shows you why. It takes an expression like x² + 8x, draws it as real areas, and rearranges the pieces into an almost-complete square so you can see exactly where the missing number comes from.
Use the sliders to choose the coefficient of x and an optional constant, then step through the five stages. It works for teachers projecting to a class and for students revising on their own.
Watch an algebraic expression turn into an almost-complete square — and see exactly where the missing number comes from.
Completed-square form
How to use the visualiser
Choose a coefficient of x between 1 and 12, then click through the steps. At each stage, the diagram and the explanation panel update together. When you reach the missing corner, try the quick check question before revealing the answer — working it out yourself is where the learning happens.
Two things worth trying once you've been through the steps. First, drag the coefficient slider while you're on the final step and watch the corner square grow and shrink as the equation updates — you'll see why a bigger coefficient means subtracting a bigger number. Second, set the coefficient to 6 and the constant to 9, and see what happens to the answer.
What is completing the square?
Completing the square means rewriting a quadratic expression like x² + 6x + 2 in the form (x + a)² + b. It's called "completing the square" for a genuinely geometric reason: if you draw x² + 6x as areas, you get a square and a rectangle that can be rearranged into a larger square with one corner missing. Filling in that corner — completing the square — is what the algebra is really doing.
Why do you halve the coefficient of x?
This is the question the visualiser answers best. The 6x rectangle has to be split into two equal halves — one placed beside the x² square and one underneath — to build up the shape of a bigger square. Each half is 3x, which is why the answer contains (x + 3) and not (x + 6). Halving isn't a rule to memorise; it's a consequence of needing two matching sides.
Why do you subtract the number at the end?
Once the two halves are in place, the shape is a square with one corner missing. That corner measures 3 by 3, so filling it adds an area of 9 that was never in the original expression. To keep the expression equal to what we started with, we take the 9 straight back off:
x² + 6x = (x + 3)² − 9
If the expression has a constant, it simply waits until the end and combines with the subtracted number:
x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7
Worked example with an odd coefficient
Odd coefficients work exactly the same way — the numbers just aren't whole. For x² + 7x, halving gives 3.5 and squaring gives 12.25:
x² + 7x = (x + 3.5)² − 12.25
In exam answers you'll usually write this with fractions: (x + 7/2)² − 49/4. Try setting the slider to 7 and stepping through — the diagram doesn't care whether the corner is a whole number.
Why completing the square matters at GCSE and beyond
Completing the square appears on higher-tier GCSE papers in its own right, but it's also the key to two bigger ideas: finding the turning point of a quadratic graph (the completed-square form hands you the coordinates directly), and solving quadratic equations that don't factorise. It's also where the quadratic formula comes from — the formula is just completing the square done once, in general, with letters.
FAQs
What does completing the square mean in maths?
It means rewriting a quadratic expression such as x² + bx + c in the form (x + a)² + k. The name comes from the area model: the expression literally forms a square with a missing corner, and the method fills it in.
How do you complete the square step by step?
Halve the coefficient of x to get the number inside the bracket. Square that number and subtract it outside the bracket. Finally, combine the subtracted number with any constant in the original expression.
Why do you halve the coefficient of x when completing the square?
Because the bx rectangle must be split into two equal pieces to build two sides of a larger square. Each piece has area (b/2)x, so the side of the completed square is x + b/2.
Does completing the square work with odd numbers?
Yes. Halving an odd coefficient gives a decimal or fraction — for example, x² + 5x = (x + 2.5)² − 6.25, or equivalently (x + 5/2)² − 25/4. The method is identical.
What is completing the square used for?
Finding the turning point (vertex) of a quadratic graph, solving quadratic equations that don't factorise, and deriving the quadratic formula. It also appears in circle equations at A Level.