Question Number 191430 by TechnicalDev last updated on 24/Apr/23
$${factorise}\:\mathrm{4}{x}^{\mathrm{4}} +\mathrm{81} \\ $$
Answered by som(math1967) last updated on 24/Apr/23
$$\mathrm{4}{x}^{\mathrm{4}} +\mathrm{81} \\ $$$$=\left(\mathrm{2}{x}^{\mathrm{2}} \right)^{\mathrm{2}} +\left(\mathrm{9}\right)^{\mathrm{2}} \\ $$$$=\left(\mathrm{2}{x}^{\mathrm{2}} +\mathrm{9}\right)^{\mathrm{2}} −\mathrm{2}.\mathrm{2}{x}^{\mathrm{2}} .\mathrm{9} \\ $$$$=\left(\mathrm{2}{x}^{\mathrm{2}} +\mathrm{9}\right)^{\mathrm{2}} −\left(\mathrm{6}{x}\right)^{\mathrm{2}} \\ $$$$=\left(\mathrm{2}{x}^{\mathrm{2}} +\mathrm{9}+\mathrm{6}{x}\right)\left(\mathrm{2}{x}^{\mathrm{2}} +\mathrm{9}−\mathrm{6}{x}\right) \\ $$$$=\left(\mathrm{2}{x}^{\mathrm{2}} +\mathrm{6}{x}+\mathrm{9}\right)\left(\mathrm{2}{x}^{\mathrm{2}} −\mathrm{6}{x}+\mathrm{9}\right) \\ $$
Answered by manxsol last updated on 24/Apr/23
$$\left(\mathrm{2}{x}^{\mathrm{2}} \right)^{\mathrm{2}} +\left(\mathrm{9}\right)^{\mathrm{2}} +\mathrm{2}\left(\mathrm{2}{x}^{\mathrm{2}} \right)\left(\mathrm{9}\right)−\left(\mathrm{6}{x}\right)^{\mathrm{2}} \\ $$$$\left(\mathrm{2}{x}^{\mathrm{2}} +\mathrm{9}\right)^{\mathrm{2}} −\left(\mathrm{6}{x}\right)^{\mathrm{2}} \\ $$$${fact}\:{real} \\ $$$$\left(\mathrm{2}{x}^{\mathrm{2}} −\mathrm{6}{x}+\mathrm{9}\right)\left(\mathrm{2}{x}^{\mathrm{2}} +\mathrm{6}{x}+\mathrm{9}\right) \\ $$$$\Delta=\mathrm{36}−\mathrm{4}\left(\mathrm{2}\right)\left(\mathrm{9}\right)=−\mathrm{36} \\ $$$${fact}\:{complex} \\ $$$$\frac{\mathrm{6}+\mathrm{6}{i}}{\mathrm{4}}=\frac{\mathrm{3}+\mathrm{3}{i}}{\mathrm{2}} \\ $$$$\left({x}−\frac{\mathrm{3}+\mathrm{3}{i}}{\mathrm{2}}\right)\left({x}−\frac{\mathrm{3}−\mathrm{3}{i}}{\mathrm{2}}\right)\left({x}−\frac{−\mathrm{3}+\mathrm{3}{i}}{\mathrm{2}}\right)\left({x}−\frac{−\mathrm{3}−\mathrm{3}{i}}{\mathrm{2}}\right) \\ $$$$ \\ $$
Answered by namphamduc last updated on 24/Apr/23
$$\mathrm{4}{x}^{\mathrm{4}} +\mathrm{81}=\left(\mathrm{2}{x}^{\mathrm{2}} \right)^{\mathrm{2}} +\mathrm{2}.\mathrm{2}{x}^{\mathrm{2}} .\mathrm{9}+\mathrm{81}−\mathrm{36}{x}^{\mathrm{2}} \\ $$$$=\left(\mathrm{2}{x}^{\mathrm{2}} +\mathrm{9}\right)^{\mathrm{2}} −\left(\mathrm{6}{x}\right)^{\mathrm{2}} =\left(\mathrm{2}{x}^{\mathrm{2}} −\mathrm{6}{x}+\mathrm{9}\right)\left(\mathrm{2}{x}^{\mathrm{2}} +\mathrm{6}{x}+\mathrm{9}\right) \\ $$