if-and-are-the-roots-of-4x-2-6x-1-00-0-find-3-3-

Question Number 63399 by smartsmith459@gmail.com last updated on 03/Jul/19

$${if}\:\alpha\:{and}\:\beta\:{are}\:{the}\:{roots}\:{of}\:\mathrm{4}{x}^{\mathrm{2}\:} −\mathrm{6}{x}+\mathrm{1}===\mathrm{00}====================== \\ $$$$=\mathrm{0}.\:{find}\:\alpha^{\mathrm{3}} −\beta^{\mathrm{3}} . \\ $$

Commented by mr W last updated on 03/Jul/19

$$\alpha+\beta=\frac{\mathrm{6}}{\mathrm{4}}=\frac{\mathrm{3}}{\mathrm{2}} \\ $$$$\alpha\beta=\frac{\mathrm{1}}{\mathrm{4}} \\ $$$$ \\ $$$$\left(\alpha−\beta\right)^{\mathrm{2}} =\left(\alpha+\beta\right)^{\mathrm{2}} −\mathrm{4}\alpha\beta \\ $$$$\Rightarrow\left(\alpha−\beta\right)=\pm\sqrt{\left(\alpha+\beta\right)^{\mathrm{2}} −\mathrm{4}\alpha\beta} \\ $$$$ \\ $$$$\alpha^{\mathrm{3}} −\beta^{\mathrm{3}} \\ $$$$=\left(\alpha−\beta\right)\left(\alpha^{\mathrm{2}} +\beta^{\mathrm{2}} +\alpha\beta\right) \\ $$$$=\left(\alpha−\beta\right)\left[\left(\alpha+\beta\right)^{\mathrm{2}} −\alpha\beta\right] \\ $$$$=\pm\sqrt{\left(\alpha+\beta\right)^{\mathrm{2}} −\mathrm{4}\alpha\beta}\left[\left(\alpha+\beta\right)^{\mathrm{2}} −\alpha\beta\right] \\ $$$$=\pm\sqrt{\frac{\mathrm{9}}{\mathrm{4}}−\mathrm{1}}\left[\frac{\mathrm{9}}{\mathrm{4}}−\frac{\mathrm{1}}{\mathrm{4}}\right] \\ $$$$=\pm\mathrm{2}\sqrt{\frac{\mathrm{5}}{\mathrm{4}}} \\ $$$$=\pm\sqrt{\mathrm{5}} \\ $$

Answered by MJS last updated on 03/Jul/19

$$\mathrm{4}{x}^{\mathrm{2}} −\mathrm{6}{x}+\mathrm{1}=\mathrm{0}\:\Rightarrow\:{x}=\frac{\mathrm{3}}{\mathrm{4}}\pm\frac{\sqrt{\mathrm{5}}}{\mathrm{4}} \\ $$$$\mathrm{if}\:\alpha<\beta\:\Rightarrow\:\alpha=\frac{\mathrm{3}}{\mathrm{4}}−\frac{\sqrt{\mathrm{5}}}{\mathrm{4}}\wedge\beta=\frac{\mathrm{3}}{\mathrm{4}}+\frac{\sqrt{\mathrm{5}}}{\mathrm{4}} \\ $$$$\Rightarrow\:\alpha^{\mathrm{3}} −\beta^{\mathrm{3}} =−\sqrt{\mathrm{5}} \\ $$

Terms of Service
Privacy Policy
Contact: info@tinkutara.com