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Question Number 166958 by cortano1 last updated on 03/Mar/22

$$\mathrm{If}\mathrm{polynomial}{\mathrm{x}}^3-{9\mathrm{x}}^2+11\mathrm{x}-1=0$$$$\mathrm{have}\mathrm{the}\mathrm{roots}\mathrm{are}\mathrm{a},\mathrm{b}\mathrm{an}\mathrm{c}.$$$$\mathrm{Given}\mathrm{\Delta }=\sqrt{\mathrm{a}}+\sqrt{\mathrm{b}}+\sqrt{\mathrm{c}}\mathrm{then}$$$${\mathrm{\Delta }}^4-18{\mathrm{\Delta }}^2-8\mathrm{\Delta }=?$$$$$$

Commented by MJS_new last updated on 03/Mar/22

$$\mathrm{the}\mathrm{answer}\mathrm{is}-37$$$$$$$$x^3-9x^2+11x-1=0$$$$\mathrm{let}x=p+3$$$$p^3-16p-22=0$$$$$$$${\mathrm{\Delta }}^4-18{\mathrm{\Delta }}^2-8\mathrm{\Delta }-\chi =0$$$$({\mathrm{\Delta }}^2-\alpha \mathrm{\Delta }-\beta )({\mathrm{\Delta }}^2+\alpha \mathrm{\Delta }-\gamma )=0$$$$\beta =-\frac{{\alpha }^2}{2}+\frac{4}{\alpha }+9\wedge \gamma =-\frac{{\alpha }^2}{2}-\frac{4}{\alpha }+9$$$$\alpha =\sqrt{q+12}$$$$q^3+4(\chi -27)q+16(3\chi +23)=0$$$$\mathrm{if}\mathrm{we}\mathrm{let}\chi =-37\Rightarrow q=4p$$$$$$$$\mathrm{but}\mathrm{someone}\mathrm{else}\mathrm{please}\mathrm{explain}\mathrm{this};\mathrm{I}{\mathrm{don}}^{\prime }\mathrm{t}$$$$\mathrm{fully}\mathrm{understand}\mathrm{what}\mathrm{I}\mathrm{did}...$$

Answered by mr W last updated on 03/Mar/22

$$a+b+c=9$$$$ab+bc+ca=11$$$$abc=1$$$$$$$$\mathrm{\Delta }=\sqrt{a}+\sqrt{b}+\sqrt{c}$$$$$$$${\mathrm{\Delta }}^2=a+b+c+2(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})$$$${\mathrm{\Delta }}^2=9+2(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})$$$$$$$${\mathrm{\Delta }}^4=81+36(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})+4(ab+bc+ca+2\sqrt{a^{2}bc}+2\sqrt{{ab}^{2}c}+2\sqrt{{abc}^2})$$$${\mathrm{\Delta }}^4=81+36(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})+4(11+2\sqrt{a}+2\sqrt{b}+2\sqrt{c})$$$${\mathrm{\Delta }}^4=125+36(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})+8(\sqrt{a}+\sqrt{b}+\sqrt{c})$$$$$$$${\mathrm{\Delta }}^4-18{\mathrm{\Delta }}^2-8\mathrm{\Delta }$$$$=125+36(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})+8(\sqrt{a}+\sqrt{b}+\sqrt{c})$$$$-18\times 9-18\times 2(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})$$$$-8\times (\sqrt{a}+\sqrt{b}+\sqrt{c})$$$$=125-18\times 9$$$$=-37$$

Answered by null last updated on 03/Mar/22

$$upsilon$$

Answered by null last updated on 04/Mar/22

$$\mathrm{upsilon}=-37$$$$\upsilon =-37$$

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