Where the Formula Comes From

The motion: “The quadratic formula is a tool to be memorised; understanding its derivation adds nothing to solving equations.”

Your opponent: Muhammad al-Khwarizmi · Course: Pearson Edexcel International GCSE Mathematics (4MA1) · Assessment objectives: AO1, AO2

Completing the square is his. He wrote it down around 820, and he drew the square to prove it.

What you are asked to argue

Defeat the motion by deriving the formula from completing the square, and by showing what the derivation lets you do that memorising does not.

Completing the square on ax² + bx + c = 0
The quadratic formula and the discriminant b² − 4ac
What the discriminant tells you about the number of roots
The connection between completing the square and the turning point of the parabola

Disagreeing with the figure is not an error. Any of these can score full marks if argued well:

That memorising is sufficient for the exam and understanding is valuable for other reasons — precise and defensible if argued that way
That the derivation matters chiefly because it generalises to other completions, which is Lovelace's own point

Wrong: You can always divide through by a without comment.
The correction: It is valid precisely because a ≠ 0 — which is part of what makes the equation quadratic. Naming that condition is part of the justification.
Wrong: The discriminant is a separate rule to learn.
The correction: It falls out of the derivation: it is the quantity under the root, so its sign decides how many real roots exist. Nothing separate needs to be learned.

A complete derivation with the a ≠ 0 condition named, and at least one demonstrated advantage of understanding over recall.

Read the life before you argue with it — the case you are about to meet was built by a real person, over a real career:

The life of Muhammad al-Khwarizmi — Who Named Algebra And The Algorithm · Mathematician & Astronomer · c. 780–850