ax² + bx + c = 0?
In the previous lesson you learned to complete the square on specific quadratics — turning x² + 6x + 5 into (x + 3)² − 4, and reading off the vertex.
This lesson does something both simple and profound: it applies that exact same technique not to a specific quadratic, but to the general quadratic equation
ax² + bx + c = 0
where a, b, and c stand for any numbers (with a ≠ 0). When you complete the square on this general equation and solve for x, what falls out the other end is the quadratic formula itself:
x = (−b ± √(b² − 4ac)) / 2a
This is the central idea of the lesson: the quadratic formula is not a separate fact to memorise — it is what completing the square looks like when you do it once, in general, for every quadratic at the same time. Every time you "use the formula", you are reusing a single completed square that someone did long ago, so you don't have to do it again each time.
The interactive in The Designed World walks through the derivation step by step. Before you reach for it, see how far you can get on your own — you genuinely have all the tools you need.
The quadratic formula has one of the longest histories of any single result in mathematics — and for most of that history, there was no formula, only methods.
The Babylonians, around 2000 BCE, could already solve quadratic problems — working in base 60, on clay tablets, using a recipe that is essentially completing the square, though they would never have written it as a formula. They had no symbol for an unknown, no concept of a negative number, and certainly no ±. They had a procedure: do this, then this, then this, and you get the answer.
For nearly four thousand years, that's how quadratics were solved — as procedures, described in words, often tied to specific geometric pictures. Al-Khwarizmi in 9th-century Baghdad, whose completing-the-square method we met in the last lesson, still wrote everything in prose. The Indian mathematician Brahmagupta (7th century) came remarkably close to a general rule and, importantly, was willing to work with negative numbers — a crucial step, because the ± in the modern formula only makes sense once you accept that a quadratic can have two solutions.
The formula as we write it — compact, symbolic, with letters standing for any coefficients — could not exist until symbolic algebra itself was invented. That came gradually, with mathematicians like François Viète (16th century) introducing letters for unknowns and constants, and René Descartes (17th century) giving us much of the notation we still use. Only then could the accumulated wisdom of four thousand years be compressed into a single line.
There's a lovely lesson hidden here about the power of notation. The Babylonians, al-Khwarizmi, and Brahmagupta were not less clever than us — in many ways they were extraordinary. What they lacked was a language compact enough to say "for any a, b, c..." in a single breath. The formula you're about to derive is not just a piece of mathematics; it's a monument to the slow human invention of a language good enough to hold it. When you write x = (−b ± √(b² − 4ac)) / 2a, you are using a sentence that took civilisations to learn how to write.
Choose one of the following. There is no single right answer.
ax² + bx + c = 0 to the formula, without looking. Then check it. Where did you get stuck? That sticky step is the one worth understanding most deeply.2a is on the bottom and where the b² − 4ac under the root "comes from". Write it as a short dialogue, or record it as a voice note. Teaching is the strongest test of understanding.x² + 6x + 5" to "completing the square on ax² + bx + c" is the leap from solving one problem to solving all problems of that kind at once. Write a paragraph about another moment — in maths or in life — where solving the general case once saves you from solving the specific case forever.Deriving a result you already "knew" transforms it from something you trust into something you own.
This lesson's "designed object" is the derivation itself — a small piece of mathematical engineering, built one step at a time. Each step has a goal (what we're trying to achieve) and a move (the algebra that achieves it). Before you reveal each step's reasoning, pause and try to predict it: what would I do next, and why?
Work through it below. Try each step in your head (or on paper) first, then reveal the thinking, then advance.
Starting point: the general quadratic equation, with a ≠ 0.
And there it is — derived, not memorised:
Every quadratic you ever solve with "the formula" is reusing this single completed square.
Notice the expression that appears under the square root: b² − 4ac. Keep it in mind — in the next lesson, that single expression turns out to decide everything about how many solutions a quadratic has. It even has a name: the discriminant.
Take a breath. There's a particular feeling that comes with deriving something you'd previously only memorised — a kind of "oh, that's all it is" relief, mixed sometimes with a little indignation that nobody showed you sooner. Some questions to sit with:
± appears, or the tidying of the right-hand side into a single fraction.) That step is worth a second visit — not because you got it wrong, but because that's where the real understanding lives.If you'd like a short reset before the Final Task: close your eyes for a moment and try to "replay" the derivation as a film — equation transforming into equation. Don't worry if it's blurry; the blurry parts are just showing you where to look again.
Produce a short piece of work (1–2 sides of A4, digital or handwritten) titled The Derivation, Owned, containing all of the following:
ax² + bx + c = 0 and reach x = (−b ± √(b² − 4ac)) / 2a. Annotate each step with a few words saying what you did and why.2x² + 7x − 4 = 0, then check your two answers by substituting them back into the original equation.b² − 4ac wherever it appears in your derivation, and write one sentence predicting why this expression might be important. (You'll meet it properly as "the discriminant" in the next lesson.)| Strand | What we're looking for |
|---|---|
| Conceptual Understanding | The annotations and the "hard step" explanation show genuine understanding of why each move works — especially the appearance of the ± and the role of dividing by a. |
| Fluency & Accuracy | The derivation is algebraically correct throughout; the verification solves 2x² + 7x − 4 = 0 correctly and the check is carried out. |
| Application to Problems | The derived formula is applied correctly to a concrete equation, and the answers are verified by substitution. |
| Independence & Reflection | The derivation is genuinely reconstructed (not copied); the reflection shows honest thought about the difference between trusting and owning a result. |
Take whatever form for this work 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.
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