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Question Number 205051 by mr W last updated on 06/Mar/24

$$\mathrm{Find}\mathrm{all}\mathrm{values}\mathrm{of}\mathrm{k}\mathrm{such}\mathrm{that}\mathrm{the}$$$$\mathrm{expr}e\mathrm{ssion}{\mathrm{x}}^3+{\mathrm{kx}}^2-7\mathrm{x}+6\mathrm{can}\mathrm{be}$$$$\mathrm{re}s\mathrm{olved}\mathrm{into}\mathrm{three}\mathrm{linear}\mathrm{real}\mathrm{factors}.$$

Answered by Frix last updated on 06/Mar/24

$$k<c\mathrm{with}c\mathrm{being}\mathrm{the}\mathrm{real}\mathrm{solution}\mathrm{of}$$$$c^3-\frac{49c^2}{24}+\frac{63c}{2}-\frac{50}{3}=0$$$$c\approx .543134153288$$

Commented by mr W last updated on 06/Mar/24

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Answered by mr W last updated on 07/Mar/24

$$f\left(x\right)=x^3+{kx}^2-7x+6$$$$f^{\prime }\left(x\right)=3x^2+2kx-7$$$$atx=p:f\left(p\right)=0,f^{\prime }\left(p\right)=0$$$$p^3+{kp}^2-7p+6=0...\left(i\right)$$$$3p^2+2kp-7=0...\left(ii\right)$$$$p^3+7p-12=0$$$$p=\sqrt[3]{\frac{\sqrt{3945}}{9}+6}-\sqrt[3]{\frac{\sqrt{3945}}{9}-6}$$$$k=\frac{1}{2}(\frac{7}{p}-3p)\approx 0.54313415$$

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