Al-Khwarizmi · Mathematics
The geometric heart of Al-Khwarizmi’s method - how literally building a square out of an equation turns a recipe into a proof, and reveals why the quadratic formula works.
Return to Al-Khwarizmi’s famous equation, ‘a square and ten roots are equal to thirty-nine’ - in our notation, x² + 10x = 39. His recipe solves it: halve the ten (get 5), square that (get 25), add to thirty-nine (get 64), take the square root (get 8), subtract the five (get 3). The answer, 3, is correct. But why? Where do these particular steps come from? Why halve the roots and not divide by three? Why square the half? Why add it to the other side? A recipe you cannot justify is a magic spell, not mathematics - and Al-Khwarizmi, schooled in the Greek tradition of proof, would not rest on a spell. His answer is one of the most beautiful ideas in elementary mathematics: the recipe is the arithmetic shadow of a picture. If you draw the equation as an actual geometric figure - a square with rectangles attached - then completing that figure into a larger, perfect square forces exactly these steps upon you. The algebra is justified by the geometry. This lesson is about seeing the picture behind the formula.
Here is Al-Khwarizmi’s construction, in the version that is easiest to picture. Draw a square whose side is the unknown x; its area is x². Now the ‘ten roots’ - the 10x - must be distributed around this square. Split the ten into two fives, and attach a rectangle of width 5 to two adjacent sides of the square (each rectangle has area 5x, and the two together have area 10x, the ten roots). The figure is now an L-shape: the original square plus two rectangles along two of its sides. Its total area is x² + 10x, which the equation tells us equals 39. But the L-shape has a square notch missing in its outer corner - a little square whose side is 5 (the width of the rectangles) and whose area is therefore 5 × 5 = 25. Add that missing 25 to complete the figure into a single large square. The large square has area 39 + 25 = 64, and since its side is x + 5 (the original side plus the rectangle width), we have (x + 5)² = 64. Take the root: x + 5 = 8, so x = 3. Every number in the recipe - the halving to get 5, the squaring to get 25, the adding to get 64, the rooting to get 8, the subtracting to get 3 - appears in the picture as an actual length or area.
The deepest payoff of Al-Khwarizmi’s geometric method is that, generalised, it is the derivation of the quadratic formula - the formula every student memorises but few are shown the reason for. Take the general equation x² + bx = c (a square and some roots equal a number). Completing the square geometrically, as we have seen, adds (b/2)² to both sides: the left becomes a perfect square (x + b/2)² and the right becomes c + (b/2)². Taking the square root of both sides gives x + b/2 = √(c + (b/2)²), and subtracting b/2 isolates the unknown: x = −b/2 + √((b/2)² + c). For the full quadratic ax² + bx + c = 0, the same completing-the-square procedure - divide through by a, move the constant over, add the square of half the new middle coefficient - yields the celebrated formula x = (−b ± √(b² − 4ac)) / 2a. Every quadratic formula in every textbook is Al-Khwarizmi’s figure, written in symbols he never had. The mysterious ‘b² − 4ac under the square root’ is the area of a completed square; the ‘minus b over 2a’ is the subtraction of the rectangle’s half-width. To complete the square is to derive the formula - and to derive the formula is to draw Al-Khwarizmi’s picture in algebra.
This is the opening of the lesson. The rest — the dialogue, the primary source, and the recall — is in the app.
You learned how Al-Khwarizmi proves his solution recipes geometrically by ‘completing the square’ - attaching rectangles to a square and filling in the missing corners. Explain how this geometric construction justifies the algebraic procedure and connects to the quadratic formula.
Leads to Euclid.
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