If the equation $x^4+a x^3+b x^2+c x+d=0$ has three equal roots, then that root is

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Aicharya · Accepted Answer

Let $\alpha, \alpha, \alpha, \beta$ be the roots of given equation $ x^4+a x^3+b x^2+c x+d=0 $ Hence $3 \alpha+\beta=-a, \quad 3 \alpha(\alpha+\beta)=b$ $ \alpha^2(\alpha+3 \beta)=-c, \quad \alpha^3 \beta=d $ Now consider $6 c-a b=\alpha\left(3 \alpha^2-6 \alpha \beta+3 \beta^2\right)....(1)$ also consider $3 a^2-8 b=\left(3 \alpha^2-6 \alpha \beta+3 \beta^2\right)....(2)$ dividing equation (i) by equation (2), $ \Rightarrow \alpha=\frac{6 c-a b}{3 a^2-8 b} $

If the equation $x^4+a x^3+b x^2+c x+d=0$ has three equal roots, then that root is

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