if-a-and-b-are-real-numbers-determine-a-necessary-and-sufficient-condition-so-as-the-equation-x-2-a-b-x-a-b-to-have-three-real-distinct-roots-

Question Number 154980 by mathdanisur last updated on 23/Sep/21

if \boldsymbol a and \boldsymbol b are real numbers

determine a necessary and sufficient

condition so as the equation

x^{2} + \frac{a \sqrt{b}}{x} = a + b

to have three real distinct roots .

Answered by mr W last updated on 24/Sep/21

x \neq 0, b > 0

x^{3} - (a + b) x + a \sqrt{b} = 0

f o r t h r e e d i s t i n c t r o o t s :

\frac{{(a \sqrt{b})}^{2}}{4} - \frac{{(a + b)}^{3}}{27} < 0

\Rightarrow 4 {(a + b)}^{3} - 27 a^{2} b > 0

Commented by mathdanisur last updated on 24/Sep/21

Thank you \boldsymbol S er, how did you get 4 and 27

please

Commented by mathdanisur last updated on 24/Sep/21

Thank you \boldsymbol S er

Commented by mr W last updated on 24/Sep/21

i j u s t a p p l i e d t h e {c a r d a n o}^{'} s f o r m u l a

f o r c u b i c e q u a t i o n .

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