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January 5th, 2019, 08:46 AM   #1
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Red face Prove general solution for deppresed cubic equation y^3+py+q=0 given a discriminant

I've been given the following problem and I can't seem to figure out the solution, would appreciate any help and direction to solution !

Let's look at the equation : y3 + py + q = 0 (*)
We shall define Delta (or d in short) d = 4p3 + 27q2

Prove that :
a) If d > 0 , then (*) has a single solution .

b) If d = 0 , and at least one of the coefficients (p , q) =/= 0, then (*) has 2 solutions .

c) If d < 0 , then (*) has 3 solutions .


Thank you very much in advance !
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January 5th, 2019, 01:10 PM   #2
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You could write the equation as $(y-a)(y-b)(y-c)=0$ Compute p and q, noting that the coefficient of $y^2$ is 0 and then set up d. Then look at what happens to a, b, and c.
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January 5th, 2019, 01:16 PM   #3
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Thank you very much !
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