if-and-are-the-positive-roots-of-the-eqn-x-2-px-q-0-find-the-sum-M-1-4-1-4-

Question Number 171922 by Mikenice last updated on 21/Jun/22

i f α a n d β a r e t h e p o s i t i v e r o o t s o f t h e e q n

x^{2} + p x + q = 0, f i n d t h e s u m M = \sqrt[4]{α} + \sqrt[4]{β}

Commented by kaivan.ahmadi last updated on 22/Jun/22

w e n o w S = α + β = - p, P = α β = q

M^{2} = \sqrt{α} + \sqrt{β} + 2 \sqrt[4]{α β}

o n t h e o t h e r e h a n d i f K = \sqrt{α} + \sqrt{β} t h e n

K^{2} = α + β + 2 \sqrt{α β} = S + 2 \sqrt{P}

\Rightarrow K = \sqrt{S + 2 \sqrt{P}} = \sqrt{- p + 2 \sqrt{q}}

s o w e h a v e

M^{2} = K + 2 \sqrt[4]{P} = \sqrt{- p + 2 \sqrt{q}} + 2 \sqrt[4]{q}

Commented by Mikenice last updated on 23/Jun/22

t h a n k s s i r