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Conjugate of a Complex Number

If one root o...

Question

If one root of the equation $(1 - m) x^{2} + l x + 1 = 0$ is double of the other and $l$ is real,find the greatest value of $m$ .

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Solution

$(1 - m) x^{2} + l x + 1 = 0 - - a - - - 2 a$
$2 a^{2} = \frac{1}{1 - m}$
$3 a = \frac{- l}{1 - m} \Rightarrow 1 - m = \frac{- l}{3 a} \Rightarrow 2 a^{2} = \frac{- 3 x}{l} l = \frac{- 3}{2 a}$
$3 a = \frac{- 3}{2 a (1 - m)} \Rightarrow m = \frac{1}{2 a^{2}} + 1$

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