Category: Algebra

Question Number 211099 by peter frank last updated on 27/Aug/24 Answered by A5T last updated on 27/Aug/24 $$\mathrm{2}^{{logx}} =\mathrm{2}^{\frac{{log}_{\mathrm{2}} {x}}{{log}_{\mathrm{2}} \mathrm{10}}} ={x}^{\frac{\mathrm{1}}{{log}_{\mathrm{2}} \mathrm{10}}} ={x}^{{log}_{\mathrm{10}} \mathrm{2}}…

Question Number 211029 by BaliramKumar last updated on 26/Aug/24 Answered by Ar Brandon last updated on 26/Aug/24 $$f\left(2\right)=5\mathrm{therefore}f\left(x\right)\mathrm{must}\mathrm{be}\mathrm{a}\mathrm{non}-\mathrm{zero}\mathrm{polynomial}.$$$$\mathrm{Let}\:{f}\left({x}\right)={a}_{\mathrm{0}} +{a}_{\mathrm{1}} {x}+{a}_{\mathrm{2}} {x}^{\mathrm{2}} +\centerdot\centerdot\centerdot+{a}_{{n}} {x}^{{n}}…

Question Number 210971 by black_mamba234 last updated on 25/Aug/24 Answered by A5T last updated on 25/Aug/24 $$2)2x_1+4y_1-2z_1=2(x_1+k)+4y_1-2(z_1+k)$$$${f}\left({x}_{\mathrm{1}}…

Question Number 210996 by RojaTaniya last updated on 25/Aug/24 Answered by Frix last updated on 26/Aug/24 $$\mathrm{Only}\mathrm{true}\mathrm{for}\mathrm{the}\mathrm{real}\mathrm{solution}>0:$$$$x^3+2x^2+5x-1=0$$$${x}^{\mathrm{3}} +\mathrm{6}{x}^{\mathrm{2}} +\mathrm{9}{x}=\mathrm{4}{x}^{\mathrm{2}}…

Question Number 210961 by RojaTaniya last updated on 24/Aug/24 Answered by A5T last updated on 24/Aug/24 $$\frac{a-1+2}{a-1}+\frac{b-1+2}{b-1}+\frac{c-1+2}{c-1}=10$$$$\Rightarrow \frac{1}{a-1}+\frac{1}{b-1}+\frac{1}{c-1}=\frac{7}{2}$$$$a+b+c=0;ab+bc+ca=\frac{m-1}{m};abc=\frac{-m-1}{m}$$$$\Rightarrow \frac{ab+bc+ca-2(a+b+c)+3}{abc-ab-bc-ac+a+b+c-1}=\frac{7}{2}$$$$\Rightarrow\frac{\frac{{m}−\mathrm{1}+\mathrm{3}{m}}{{m}}}{\frac{−{m}−\mathrm{1}+\mathrm{1}−{m}−{m}}{{m}}}=\frac{\mathrm{7}}{\mathrm{2}}\Rightarrow\frac{\mathrm{4}{m}−\mathrm{1}}{−\mathrm{3}{m}}=\frac{\mathrm{7}}{\mathrm{2}}…