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Greek Solution

The Greeks thought of a product of two things as an area. For example, ab was thought of as a rectangle with base a and height b. This idea is used in Euclid's "proof" of the quadratic equation a(a - x) = x2 (or x2 + ax - a2 = 0). Note, Euclid's "proof" was more of a construction of the positive root of the equation, followed by a verification. Euclid's Proposition 11, from Book II of Elements is:

To cut a given straight line so that the rectangle contained by the whole and one of the segments is equal to the square on the remaining segment.

Geometric figure described below.

The goal is to find a point H on the line AB so that AB × HB equals AH × AH. Letting x = AH, and a = AB, then a - x = HB, and the equation to be solved for x is a(a - x) = x2 or x2 + ax - a2 = 0. AB, or a, is the given segment. From this, a square with side a is constructed. This being square ABDC in the figure above. AC is then bisected at point E. EB is drawn, and CA is extended to a point F so that EF = EB. Next draw square FGHA. Now H is the desired point so that x = AH is the positive root of x2 + ax - a2 = 0.

Below on the left is Euclid's verification. On the right modern notation and explanations are given.

By proposition II, 6 Prop. II, 6 is basically the identity
CF × FG + AE2 = EF2 (y + z)(y - z) + z2 = y2(1)
or
(y + z)(y - z) = y2 - z2
where y = x + a/2 and z = a/2 so that
By construction EF = EB; y + z = x + a and y - z = x.
thus This gives us
CF × FG + AE2 = EB2 (a + x)(x) + (a/2)2 = (x + a/2)2 in (1) above.
By Pythagorean Theorem, By Pythagorean Theorem,
CF × FG + AE2 = AB2 + AE2 (a + x)(x) + (a/2)2 = a2 + (a/2)2
-AE2-AE2 -(a/2)2-(a/2)2


CF × FG=AB2 (a + x)(x)=a2
-AHKC-AHKC -ax-ax


AH2 = DB × HB x2 = a(a - x)
or
AH2 = AB × HB (2)

Thus H is the required point so that AH, or x, satisfies (2) on the left above.

You can try this by letting AB = a = 3 to get x2 + 3x - 9 = 0. If you do the above construction, you should find that AX = x = 1.2426, which agrees with the positive root, solution to x<sup>2</sup> + 3x - 9 = 0 from the quadratic formula.