Question Number 5935 by Master Tria last updated on 05/Jun/16
$$\mathrm{If}\:{a},\:{b},\:{c},\:{x},\:{y},\:{z}\:\in\:{R}\:\mathrm{be}\:\mathrm{such}\:\mathrm{that}\: \\ $$$$\left({a}+{b}+{c}\right)^{\mathrm{2}} =\mathrm{3}\left({ab}+{bc}+{ca}−{x}^{\mathrm{2}} −{y}^{\mathrm{2}} −{z}^{\mathrm{2}} \right), \\ $$$$\mathrm{then} \\ $$
Commented by prakash jain last updated on 07/Jun/16
$$\mathrm{Is}\:\mathrm{it}\:\mathrm{an}\:\mathrm{objective}\:\mathrm{question}\:\mathrm{with}\:\mathrm{opions}? \\ $$
Answered by Yozzii last updated on 05/Jun/16
$$\left({a}+{b}+{c}\right)^{\mathrm{2}} =\mathrm{3}\left({ab}+{bc}+{ca}−{x}^{\mathrm{2}} −{y}^{\mathrm{2}} −{z}^{\mathrm{2}} \right) \\ $$$$\left({a}+{b}+{c}\right)^{\mathrm{2}} −\mathrm{2}\left({ab}+{bc}+{ca}\right)={ab}+{bc}+{ca}−\mathrm{3}\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} \right) \\ $$$${a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} ={ab}+{bc}+{ca}−\mathrm{3}\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} \right) \\ $$$${a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} −{ab}−{bc}−{ca}=−\mathrm{3}\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} \right) \\ $$$$\left({a}+{b}+{c}\right)\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} −{ab}−{bc}−{ca}\right)+\mathrm{3}{abc}=\mathrm{3}{abc}−\mathrm{3}\left({a}+{b}+{c}\right)\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} \right) \\ $$$${a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} =\mathrm{3}\left({abc}−\left({a}+{b}+{c}\right)\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} \right)\right) \\ $$$$ \\ $$
Commented by Yozzii last updated on 05/Jun/16
$${I}'{m}\:{unsure}\:{of}\:{what}\:{is}\:{being}\:{asked} \\ $$$${of}\:{the}\:{question}. \\ $$