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If the equation ax3+3bx2+3cx+d=0 has two equal roots,show that it must be equal to bc-ad2(ac-b2).
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2 Answers
2305Let the given root be y.
If the equation has 2 equal roots,
f(y)=f'(y)=0
From the above 2 condtions we get,
ay^3+3by^2+3cy+d=0....(1)
ay^2+2by+c=0....(2)
(1)-(2)\times y
\rightarrow by^2+2cy+d=0.......(3)
Solving equations 2 and 3 by determinant method,
\frac{y^2}{2c^2-2bd}=\frac{-y}{bc-ad}=\frac{1}{2b^2-2ac}
\rightarrow y =\frac{bc-ad}{2(ac-b^2)}
