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Question

Find the condition that the equation ax3+3bx2+3cx+d=0 may have two roots equal.

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Solution

In this case the equations f(x)=0, and f(x)=0, that is
ax3+3bx2+3cx+d=0....(1),
ax2+2bx+c=0....(2)
must have a common root, and the condition required will be obtained by eliminating x between these two equations.
By combining (1) and (2), we have
bx2+2cx+d=0....(3).
From (2) and (3), we obtain
x22(bdc2)=xbcad=12(acb2);
thus the required condition is
(bcad)2=4(acb2)(bdc2).

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