Question Number 191798 by Abdullahrussell last updated on 30/Apr/23
Commented by Frix last updated on 30/Apr/23
$$\mathrm{Qu}\:\mathrm{191753};\:\mathrm{just}\:\mathrm{add}\:−\mathrm{3}+\mathrm{24}=\mathrm{21} \\ $$
Commented by BaliramKumar last updated on 01/May/23
$$−\mathrm{3}×\mathrm{24}\:=\:−\mathrm{72}\:\:\: \\ $$👇
Answered by mr W last updated on 30/Apr/23
$${x}^{\mathrm{2}} {y}+{y}^{\mathrm{2}} {z}+{z}^{\mathrm{2}} {x}+{xy}^{\mathrm{2}} +{yz}^{\mathrm{2}} +{zx}^{\mathrm{2}} \\ $$$$={xy}\left({x}+{y}\right)+{yz}\left({y}+{z}\right)+{zx}\left({z}+{x}\right) \\ $$$$={xy}\left({x}+{y}+{z}\right)+{yz}\left({y}+{z}+{x}\right)+{zx}\left({z}+{x}+{y}\right)−\mathrm{3}{xyz} \\ $$$$=\left({x}+{y}+{z}\right)\left({xy}+{yz}+{zx}\right)−\mathrm{3}{xyz} \\ $$$$=\mathrm{6}×\mathrm{3}−\mathrm{3}×\left(−\mathrm{1}\right) \\ $$$$=\mathrm{21}\:\checkmark \\ $$
Answered by BaliramKumar last updated on 01/May/23
$$\left({x}^{\mathrm{2}} {y}+{y}^{\mathrm{2}} {z}+{z}^{\mathrm{2}} {x}\right)\left({xy}^{\mathrm{2}} +{yz}^{\mathrm{2}} +{zx}^{\mathrm{2}} \right) \\ $$$$=\:\left\{{x}^{\mathrm{4}} {yz}+{xy}^{\mathrm{4}} {z}+{xyz}^{\mathrm{4}} \right\}+\left\{\left({xy}\right)^{\mathrm{3}} +\left({yz}\right)^{\mathrm{3}} +\left({zx}\right)^{\mathrm{3}} \right\}+\left\{\mathrm{3}\left({xyz}\right)^{\mathrm{2}} \right\}\:\:\:\:\:\:\:\:\: \\ $$$$=\:\left\{−{x}^{\mathrm{3}} −{y}^{\mathrm{3}} −{z}^{\mathrm{3}} \right\}+\left\{\left({xy}+{yz}+{zx}\right)\left(\left({xy}+{yz}+{zx}\right)^{\mathrm{2}} −\mathrm{3}\left({xy}^{\mathrm{2}} {z}+{xyz}^{\mathrm{2}} +{x}^{\mathrm{2}} {yz}\right)\right)+\mathrm{3}{x}^{\mathrm{2}} {y}^{\mathrm{2}} {z}^{\mathrm{2}} \right\}+\left\{\mathrm{3}\right\}\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$=\:−\left\{\left({x}+{y}+{z}\right)\left(\left({x}+{y}+{z}\right)^{\mathrm{2}} −\mathrm{3}\left({xy}+{yz}+{zx}\right)\right)+\mathrm{3}{xyz}\right\}+\left\{\mathrm{3}\left(\mathrm{3}^{\mathrm{2}} −\mathrm{3}\left(−{x}−{y}−{z}\right)\right)+\mathrm{3}\left(−\mathrm{1}\right)^{\mathrm{2}} \right\}\:+\left\{\:\mathrm{3}\right\}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$=\:−\left\{\mathrm{6}\left(\mathrm{6}^{\mathrm{2}} −\mathrm{3}×\mathrm{3}\right)+\mathrm{3}\left(−\mathrm{1}\right)\right\}+\left\{\mathrm{3}\left(\mathrm{9}−\mathrm{3}\left(−\mathrm{6}\right)\right)+\mathrm{3}\right\}+\left\{\mathrm{3}\right\}\:\:\:\:\:\:\:\:\:\:\: \\ $$$$=\:−\left\{\mathrm{159}\right\}+\left\{\mathrm{84}\right\}+\left\{\mathrm{3}\right\} \\ $$$$=\:\begin{array}{|c|}{−\mathrm{72}}\\\hline\end{array} \\ $$$$ \\ $$$${p}+{q}\:=\:\mathrm{21}\:\:\:\:\:\:\:\:{pq}\:=\:−\mathrm{72} \\ $$$${s}^{\mathrm{2}} −\left({p}+{q}\right){s}+\left({pq}\right)\:=\:\mathrm{0} \\ $$$${s}^{\mathrm{2}} −\mathrm{21}{s}−\mathrm{72}=\mathrm{0} \\ $$$${s}\:=\:−\mathrm{3}\:{or}\:\mathrm{24} \\ $$$$ \\ $$