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Question Number 27046 by ajfour last updated on 01/Jan/18

C o n s i d e r i n g y = x^{3} + p x + q

I f \frac{d y}{d x} ∣_{x = α} = 0 \Rightarrow α^{2} = - \frac{p}{3}

i f \frac{d (y / x)}{d x} ∣_{x = β} = 0 \Rightarrow β^{3} = \frac{q}{2}

r o o t s o f t h e c u b i c {e q}^{n} a r e :

x = {[- β^{3} \pm \sqrt{β^{6} - α^{6}}]}^{1 / 3}

- {[β^{3} \pm \sqrt{β^{6} - α^{6}}]}^{1 / 3} .

W h y s u c h a c o n n e c t i o n ?

I f e q u a t i o n i s q u a d r a t i c e v e n_

y = {a x}^{2} + b x + c

\frac{d y}{d x} ∣_{x = α} = 0 \Rightarrow α = - \frac{b}{2 a}

\frac{d (y / x)}{d x} ∣_{x = β} = 0 \Rightarrow β^{2} = \frac{c}{a}

r o o t s o f q u a d r a t i c e q . a r e :

x = α \pm \sqrt{α^{2} - β^{2}}

w h y s u c h a c o n n e c t i o n ?