If alpha,beta are the roots of ax^2 + bx + c = 0 andn alpha + beta, alpha^2 + beta^2,alpha^3 + beta^3n are in G.P. and Delta = b^2 - 4ac, then

n n (a): (alpha^2+beta^2)^2=(alpha+beta)(alpha^3+beta^3) n ((b^2-2 a c / a^2))^2=(-(b / a))((-b^3+3 a b c / a^3)) n => 4 a^2 c^2=a c b^2 => a c(b^2-4 a c)=0n nnnAs a ≠ 0 => c Delta=0

If \alpha,\beta are the roots of ax^{2} + bx + c = 0 an... | Sequence & Series (Mathematics)

\n\begin{aligned}\n& \text { (a): }\left(\alpha^2+\beta^2\right)^2=(\alpha+\beta)\left(\alpha^3+\beta^3\right) \\\n& \left(\frac{b^2-2 a c}{a^2}\right)^2=\left(-\frac{b}{a}\right)\left(\frac{-b^3+3 a b c}{a^3}\right) \\\n& \Rightarrow 4 a^2 c^2=a c b^2 \Rightarrow a c\left(b^2-4 a c\right)=0\n\end{aligned}\n

\n\nAs

a \neq 0 \Rightarrow c \Delta=0

Explanation

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