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Question Number 9874 by j.masanja06@gmail.com last updated on 11/Jan/17
if the eqution x^3 +3hx + q=0 has the  roots α,β  and γ.  find the eqution with the roots of;  (i)(1/(αβ)) + (1/(βγ)) + (1/(αγ))  (ii)(1/α^2 ) + (1/β^2 ) + (1/γ^2 )
$$\mathrm{if}\:\mathrm{the}\:\mathrm{eqution}\:\mathrm{x}^{\mathrm{3}} +\mathrm{3hx}\:+\:\mathrm{q}=\mathrm{0}\:\mathrm{has}\:\mathrm{the} \\ $$$$\mathrm{roots}\:\alpha,\beta\:\:\mathrm{and}\:\gamma. \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{eqution}\:\mathrm{with}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}; \\ $$$$\left(\mathrm{i}\right)\frac{\mathrm{1}}{\alpha\beta}\:+\:\frac{\mathrm{1}}{\beta\gamma}\:+\:\frac{\mathrm{1}}{\alpha\gamma} \\ $$$$\left(\mathrm{ii}\right)\frac{\mathrm{1}}{\alpha^{\mathrm{2}} }\:+\:\frac{\mathrm{1}}{\beta^{\mathrm{2}} }\:+\:\frac{\mathrm{1}}{\gamma^{\mathrm{2}} } \\ $$$$ \\ $$
Commented by sandy_suhendra last updated on 11/Jan/17
(i) (1/(αβ)) + (1/(βγ)) + (1/(αγ))    or   (1/(αβ))   ,  (1/(βγ))  and (1/(αγ)) ?  (ii) (1/α^2 )+(1/β^2 )+(1/γ_ ^2 )  or  (1/α^2 )  ,  (1/β^2 )  and (1/γ^2 )  ?
$$\left(\mathrm{i}\right)\:\frac{\mathrm{1}}{\alpha\beta}\:+\:\frac{\mathrm{1}}{\beta\gamma}\:+\:\frac{\mathrm{1}}{\alpha\gamma}\:\:\:\:\mathrm{or}\:\:\:\frac{\mathrm{1}}{\alpha\beta}\:\:\:,\:\:\frac{\mathrm{1}}{\beta\gamma}\:\:\mathrm{and}\:\frac{\mathrm{1}}{\alpha\gamma}\:? \\ $$$$\left(\mathrm{ii}\right)\:\frac{\mathrm{1}}{\alpha^{\mathrm{2}} }+\frac{\mathrm{1}}{\beta^{\mathrm{2}} }+\frac{\mathrm{1}}{\gamma_{} ^{\mathrm{2}} }\:\:\mathrm{or}\:\:\frac{\mathrm{1}}{\alpha^{\mathrm{2}} }\:\:,\:\:\frac{\mathrm{1}}{\beta^{\mathrm{2}} }\:\:\mathrm{and}\:\frac{\mathrm{1}}{\gamma^{\mathrm{2}} }\:\:? \\ $$
Commented by j.masanja06@gmail.com last updated on 12/Jan/17
oooh sorry sir   has a roots of ;        (i)(1/(αβ)) ,(1/(βγ))  and (1/(βγ))          (ii)(1/α^2 ),(1/β^2 ) and (1/γ^2 )
$$\mathrm{oooh}\:\mathrm{sorry}\:\mathrm{sir}\: \\ $$$$\mathrm{has}\:\mathrm{a}\:\mathrm{roots}\:\mathrm{of}\:; \\ $$$$\:\:\:\:\:\:\left(\mathrm{i}\right)\frac{\mathrm{1}}{\alpha\beta}\:,\frac{\mathrm{1}}{\beta\gamma}\:\:\mathrm{and}\:\frac{\mathrm{1}}{\beta\gamma} \\ $$$$\:\:\:\:\:\:\:\:\left(\mathrm{ii}\right)\frac{\mathrm{1}}{\alpha^{\mathrm{2}} },\frac{\mathrm{1}}{\beta^{\mathrm{2}} }\:\mathrm{and}\:\frac{\mathrm{1}}{\gamma^{\mathrm{2}} } \\ $$

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