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You are not logged in. #1 20120120 09:16:30
Completing the cube!!!hi bobbym The limit operator is just an excuse for doing something you know you can't. “It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman “Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment #2 20120120 09:25:11
Re: Completing the cube!!!We sure can as long as you change the title to the correct one. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #3 20120120 09:28:15
Re: Completing the cube!!!could you change it.i don't know how i wrote square instead of cube. The limit operator is just an excuse for doing something you know you can't. “It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman “Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment #4 20120120 09:29:40
Re: Completing the cube!!!You could try editing the first post. But I have already done it for you. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #5 20120120 09:32:01
Re: Completing the cube!!!cannot be done that way. The limit operator is just an excuse for doing something you know you can't. “It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman “Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment #6 20120120 09:37:45
Re: Completing the cube!!!Hi anonimnystefy; If then the first three terms are the first three terms of a perfect cube, namely Then you can "complete the cube" by subtracting c from both sides and adding the missing term of the cube to both sides. Recalling that you get: By taking the cube root of the left side and the three cube roots of the right side, you get: These are the roots of the cubic equation that were sought. If then proceed as follows. Set x = y + z, where y is an indeterminate and z is a function of a, b, and c, which will be found below. Then: where The first three terms of this equation in y will be those of a perfect cube iff which happens iff which cannot happen in this case, so we seemingly haven't gained anything. However, the last three terms of this equation in y will be those of a perfect cube iff that is iff where Since then and we have a true quadratic equation, called the resolvent quadratic. Now we pick z to be a root of this quadratic equation. If then any root of the GCD is also a root of the original cubic equation in x. Once you have at least one root, the problem of finding the other roots is reduced to solving a quadratic or linear equation. If then neither value of z can make f = 0, so we can assume henceforth that f is nonzero. Either root z of the quadratic will do, but we must choose one of them. We arbitrarily pick the one with a plus sign in front of the radical: Set z equal to this value in the equation for y, and divide it by f on both sides. Then the last three terms of the cubic in y are those of a perfect cube, namely: so we can complete the cube to solve it. We do this by subtracting from both sides, then adding the missing term of the cubic, to both sides, obtaining Now you have the values of y. Add z to each to get the values of x: These are the roots of the cubic equation that were sought. Example: We have a = 6, b = 9, c = 6. Then The resolvent quadratic is the cubic in y is Then one root is After a lot of simplification, you get And two other roots that he does not provide. I checked the one he has given and it is correct. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. 