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GenePeer
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[URL='http://en.wikipedia.org/wiki/Neusis_construction']Wikipedia[/URL] said:The neusis construction (from Greek neuein = 'incline towards', plural: neuseis) consists of fitting a line element of given length (a) in between two given lines (l and m), in such a way that the line element, or its extension, passes through a given point P. That is, one end of the line element has to lie on l, the other end on m, while the line element is "inclined" towards P.
[URL='http://mathworld.wolfram.com/ConstructibleNumber.html']MathWorld[/URL] said:A number which can be represented by a finite number of additions, subtractions, multiplications, divisions, and finite square root extractions of integers. Such numbers correspond to line segments which can be constructed using only straightedge and compass.
In an essence, using normal straightedge and compass construction, we can add, subtract, multiply, divide and find a square root. This means that we can find the real roots of a quadratic equation with rational coefficients because the quadratic formula consists of only these operations. The easy and fast way to do this would be http://www.concentric.net/~pvb/ALG/rightpaths.html" .
We can also deduce that we can't solve cubic equations because the cubic formula also includes finding a cubic root. However, with neusis construction we can find the real roots of cubic equations because it can find a cube root. Does anyone know how we could solve the general cubic equation using neusis? Preferably an extension to Lill's Method.
PS: This was my topic for my http://en.wikipedia.org/wiki/Extended_essay" , and I've already handed it in. This is just further (personal) research.
JPM.
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