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

$$\mathrm{if}\mathbf{a}\mathrm{and}\mathbf{b}\mathrm{are}\mathrm{real}\mathrm{numbers}$$$$\mathrm{determine}\mathrm{a}\mathrm{necessary}\mathrm{and}\mathrm{sufficient}$$$$\mathrm{condition}\mathrm{so}\mathrm{as}\mathrm{the}\mathrm{equation}$$$${\mathrm{x}}^2+\frac{\mathrm{a}\sqrt{\mathrm{b}}}{\mathrm{x}}=\mathrm{a}+\mathrm{b}$$$$\mathrm{to}\mathrm{have}\mathrm{three}\mathrm{real}\mathrm{distinct}\mathrm{roots}.$$

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

$$x\ne 0,b>0$$$$x^3-(a+b)x+a\sqrt{b}=0$$$$forthreedistinctroots:$$$$\frac{{\left(a\sqrt{b}\right)}^2}{4}-\frac{{(a+b)}^3}{27}<0$$$$\Rightarrow 4{(a+b)}^3-27a^{2}b>0$$

Commented by mathdanisur last updated on 24/Sep/21

$$\mathrm{Thank}\mathrm{you}\mathbf{S}\mathrm{er},\mathrm{how}\mathrm{did}\mathrm{you}\mathrm{get}4\mathrm{and}27$$$$\mathrm{please}$$

Commented by mathdanisur last updated on 24/Sep/21

$$\mathrm{Thank}\mathrm{you}\mathbf{S}\mathrm{er}$$

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

$$ijustappliedthe{cardano}^{\prime }sformula$$$$forcubicequation.$$

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