A quadratic function has the form
y = ax² + bx + c (where a ≠ 0)
Its graph is always a parabola — a smooth, symmetric U-shape (opening upward if a > 0, downward if a < 0). Three features matter most:
a > 0) or maximum (if a < 0).ax² + bx + c = 0.The standard form y = ax² + bx + c tells you the curve is a parabola, but it hides the vertex. There is a second way to write the same function that reveals the vertex immediately — the vertex form (or "completed-square form"):
y = a(x − h)² + k
In this form, the vertex sits exactly at (h, k) and the axis of symmetry is the line x = h. The process of rewriting ax² + bx + c into this form is called completing the square.
Here is the method for a simple case where a = 1, say y = x² + 6x + 5:
| Step | What you do | Result |
|---|---|---|
| 1 | Take half the coefficient of x | half of 6 is 3 |
| 2 | Write (x + 3)² — this gives x² + 6x + 9 | (x + 3)² |
| 3 | But (x + 3)² is 9 too big (we only had +5), so subtract the surplus | (x + 3)² − 9 + 5 |
| 4 | Simplify | (x + 3)² − 4 |
So y = x² + 6x + 5 = (x + 3)² − 4. Reading off the vertex form: the vertex is at (−3, −4) and the axis of symmetry is x = −3. We never had to plot a single point — the algebra told us where the curve sits.
Why "completing the square"? The name is literal. The expression x² + 6x is like a square of side (x + 3) with a corner missing; adding 9 "completes" it into a perfect square. The interactive in The Designed World lets you watch this happen.
Completing the square is far older than the symbolic algebra we use to write it. When the 9th-century Baghdad mathematician al-Khwarizmi wrote his foundational book on algebra (the one whose title, al-jabr, gives us the word "algebra"), he had no symbols at all — no x, no ², no equals sign. He wrote everything in words.
And yet he solved quadratics, and he did it by completing the square — literally, geometrically, with actual squares. To solve a problem equivalent to x² + 10x = 39, al-Khwarizmi imagined a square of side x, then attached four rectangles of width 10/4 to its sides, then noticed that to make the whole thing into a bigger square he needed to add four little corner squares. "Completing the square" was, for him, a picture you could draw in sand — not a manipulation of symbols. The algebra we write today is a shorthand for a piece of geometry that is more than a thousand years old.
The parabola itself has an even older and grander history. The ancient Greeks — Menaechmus in the 4th century BCE, and later Apollonius of Perga, who literally wrote the book on them — studied parabolas as conic sections: the shapes you get by slicing a cone at various angles. They had no idea that two thousand years later, Galileo would show that a thrown object follows exactly this curve, or that the curve would turn out to focus parallel rays of light to a single point (which is why satellite dishes, car headlights, and reflecting telescopes are all parabolic). A shape studied for its pure geometric beauty turned out to be written into the physics of the universe.
There is something quietly moving in this: the Greeks studied the parabola because it was elegant, with no application in mind. Centuries later it turned out to describe falling bodies, planetary motion, and the focusing of light. Mathematics done for its own sake has a way of becoming, eventually, the language the world happens to be written in.
You can try al-Khwarizmi's idea with your own hands. The modern version is even simpler than his: instead of splitting the bx term into four rectangles, we split it into two. Below is a square of side x (area x²). Drag the two green rectangles and the gold corner piece into place to complete the square — and watch where the "extra" area comes from.
Building: x² + 6x. Drag each piece into a dashed target to complete the large square.
The gold corner piece is the heart of it. Its area — (half the x-coefficient), squared — is exactly the number you add when completing the square algebraically. Seeing it as a literal missing corner is seeing why the +9 (or +(b/2)² in general) has to be there.
Choose one of the following to explore the ideas in your own way. There is no single right answer.
x² + 6x: draw a square of side x, attach two rectangles of width 3 to two of its sides, and show the little 3×3 square needed to "complete" it. Annotate it to show why the area is (x + 3)² − 9.The way you turn a mathematical idea into your own voice — a drawing, a hunt, a metaphor — is part of what you know.
Quadratics model anything where a quantity depends on the square of another: the area of a square plot as its side changes, the stopping distance of a car as its speed rises, the height of a projectile over time, the profit of a business as a function of price (too cheap and you lose money; too expensive and nobody buys; the maximum profit sits at a vertex). In every case, finding the vertex means finding the best or worst point — the maximum profit, the minimum cost, the highest point of the throw. Completing the square is how you find it without calculus.
The interactive below lets you change a, b, and c and watch the parabola respond. As you do, the standard form is converted into vertex form live, and the vertex, axis of symmetry, and roots are marked on the graph. Watch how the completed-square form reads off the vertex directly.
a = 1, b = 6, c = 5. Read off the vertex from the vertex form. Now change only c — what moves, and what stays still? Why does changing c slide the curve vertically but not change the axis of symmetry?a = 1 and change only b. Watch the axis of symmetry move. Can you find the rule connecting b to the position of the axis of symmetry? (Hint: look at the h in the vertex form.)a to a negative value. What happens to the parabola? Where is the vertex now — a maximum or a minimum?a, b, c that give a parabola with no roots (it never crosses the x-axis). What do you notice about the vertex in that case? (This is a foretaste of the next lessons.)c (the y-intercept) obvious but the vertex hidden. No single form shows everything at once. Why might that be — and what would it take to find a form that shows the roots directly?Watch how completing the square doesn't change the curve at all — it just rewrites the same function in a form that reads off the vertex directly. The standard form and the vertex form are two descriptions of one parabola.
Take a breath. Quadratics are often the first place where algebra stops being about "find x" and starts being about shape and structure. Some questions to sit with — there are no wrong answers:
a, b, c; others prefer to trust the algebra and check the picture afterwards. Which are you? Both are legitimate mathematical styles.If you'd like a short reset before the Final Task: take three slow breaths, trace a gentle U-shape in the air with your finger, and notice the turning point at the bottom. Then come back.
Produce a short illustrated portfolio (1–2 sides of A4, digital or handwritten) titled The Parabola Portfolio, containing all of the following:
y = x² + 8x + 11 and state the vertex.y = 2x² − 12x + 7 and state the vertex. (Note the leading coefficient is not 1 — factor it out of the first two terms first.)y = x² + 8x + 11, draw the parabola, marking the vertex, the axis of symmetry, and the y-intercept. (You don't need the roots precisely — just the overall shape and position.)x metres, the area is A = x(40 − 2x). Expand this, complete the square, and use the vertex to find the value of x that gives the maximum area. What is that maximum area?| Strand | What we're looking for |
|---|---|
| Conceptual Understanding | The "two forms" reflection shows genuine grasp of why each form reveals different features — not just how to convert between them. |
| Fluency & Accuracy | Both completions of the square are correct, including the trickier case where a ≠ 1; the sketch is accurate in shape and position. |
| Application to Problems | The fencing optimisation is set up and solved cleanly, using the vertex to find the maximum rather than guessing. |
| Independence & Reflection | The chosen quadratic in the "two forms" task is genuinely yours; the reflection shows honest thinking about where optimisation matters. |
Take whatever form for the portfolio feels right — handwritten and photographed, typed and illustrated, a slide, a video walk-through. The mathematics is what's being assessed; the form is yours.
Ready to build fluency? The practice companion has exam-style questions with Socratic support whenever you get stuck.
✏️ Now Practise →