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Old 2004-04-27, 11:55   #11
R.D. Silverman
 
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Nov 2003

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Quote:
Originally Posted by jinydu
Ok, I'll try to frame my question more clearly this time.

Goal: Find a solution of x^3 + 6x - 20 = 0 and express it in the simplest possible form.

Condition: Not allowed to use trial-and-error guessing of rational roots or fore-knowledge of the solution.

Hint: Applying Cardano's method gives:

x = cube root(10+sqrt(108)) + cube root(10-sqrt(108)), but this may or may not be the simplest possible way of expressing this solution.

I'm not sure I understand the difficulty. It is easy to show
that 2 = x from x = cbr(10 + sqrt(108)) + cbr(10 - sqrt(108)) = a + b

We have

a^3 + b^3 = 20
ab = -2
x = 2
x^3 = (a+b)^3 = 8

But (a+b)^3 = a^3 + b^3 + 3abx = 20 - 6x = 20 - 6*2 = 8 = 2^3

8 = 8 QED

I'm not sure what else you are looking for.
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