First slide

Theory of equations

Question

If $α and β, α and γ, α and δ$ are the roots of the equations ${ax}^{2} + 2 bx + c = 0, 2 {bx}^{2} + cx + a = 0 and {cx}^{2} + ax + 2 b = 0$ respectively, where e, b, and c are positive real numbers, then $α + α^{2} =$

Moderate

A

abc
B

a + 2b + c
C

-1
D

0

Solution

Since $α$ is root of all equations
$\begin{aligned} {aα}^{2} + 2 bα + c = 0 \\ 2 {bα}^{2} + cα + a = 0 \\ {cα}^{2} + aα + 2 b = 0 \end{aligned}$
Adding we get $(a + 2 b + c) (α^{2} + α + 1) = 0$
$\begin{aligned} a + 2 b + c \neq 0 as a, b, c > 0 \\ \Rightarrow α^{2} + α + 1 = 0 or α^{2} + α = - 1 \end{aligned}$

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