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Math Help - Math problem (cubic equation-related)

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    Math problem (cubic equation-related)

    Find the exact value of x so that the equation has exactly 2 solutions when solved for a.
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    Quote Originally Posted by Altovirator View Post
    Find the exact value of x so that the equation has exactly 2 solutions when solved for a.
    You mean, I take it, 2 distinct real solutions. In that case, the equation must be of the form (a-u)^2(a-v)= a^3- a- x= 0. for some u and v.

    Multiplying that out, we get a^3- (2u+v)a^2+ (u^2+ 2uv)a- u^2v= a^3- a- x= 0. We must have 2u+v= 0, u^2+ 2uv= -1 and u^2v= x. Solve the first two equations for u and v and then get x from the third equation.
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