Question Number 171388 by infinityaction last updated on 15/Jun/22
$${a},{b},\:{and}\:{c}\:{are}\:{positive}\:{real}\:{numbers} \\ $$$${such}\:{that} \\ $$$${a}^{\mathrm{2}} +{ab}+\frac{{b}^{\mathrm{2}} }{\mathrm{3}}\:=\:\mathrm{25}\:\:,\:\:\:\:\frac{{b}^{\mathrm{2}} }{\mathrm{3}}\:+\:{c}^{\mathrm{2}} \:=\mathrm{9} \\ $$$${and}\:{c}^{\mathrm{2}} +{ca}+{a}^{\mathrm{2}} \:=\:\mathrm{16} \\ $$$${find}\:\:{the}\:\:{value}\:{of}\:\:{ab}+\mathrm{2}{bc}+\mathrm{3}{ac} \\ $$
Answered by MJS_new last updated on 15/Jun/22
$$\mathrm{let}\:{b}={pa}\wedge{c}={qa} \\ $$$$\left(\frac{{p}^{\mathrm{2}} }{\mathrm{3}}+{p}+\mathrm{1}\right){a}^{\mathrm{2}} =\mathrm{25} \\ $$$$\left(\frac{{p}^{\mathrm{2}} }{\mathrm{3}}+{q}^{\mathrm{2}} \right){a}^{\mathrm{2}} =\mathrm{9} \\ $$$$\left({q}^{\mathrm{2}} +{q}+\mathrm{1}\right){a}^{\mathrm{2}} =\mathrm{16}\:\left(\ast\right) \\ $$$$\Rightarrow \\ $$$$\frac{{p}^{\mathrm{2}} +\mathrm{3}{p}+\mathrm{3}}{\mathrm{75}}=\frac{{p}^{\mathrm{2}} +\mathrm{3}{q}^{\mathrm{2}} }{\mathrm{27}}=\frac{{q}^{\mathrm{2}} +{q}+\mathrm{1}}{\mathrm{16}} \\ $$$$\Rightarrow \\ $$$${p}={q}\left(\mathrm{2}{q}+\mathrm{1}\right) \\ $$$${q}^{\mathrm{4}} +{q}^{\mathrm{3}} +\frac{\mathrm{37}}{\mathrm{64}}{q}^{\mathrm{2}} −\frac{\mathrm{27}}{\mathrm{64}}{q}−\frac{\mathrm{27}}{\mathrm{64}}=\mathrm{0} \\ $$$$\left({q}^{\mathrm{2}} −\frac{\mathrm{27}}{\mathrm{64}}\right)\left({q}^{\mathrm{2}} +{q}+\mathrm{1}\right)=\mathrm{0} \\ $$$$\left(\ast\right)\:\Rightarrow\:{q}^{\mathrm{2}} +{q}+\mathrm{1}\neq\mathrm{0} \\ $$$${q}=\pm\frac{\mathrm{3}\sqrt{\mathrm{3}}}{\mathrm{8}}\:\Rightarrow\:{p}=\frac{\mathrm{27}}{\mathrm{32}}\pm\frac{\mathrm{3}\sqrt{\mathrm{3}}}{\mathrm{8}} \\ $$$$\Rightarrow\:\mathrm{insert}\:\mathrm{and}\:\mathrm{solve}\:\mathrm{for}\:{a} \\ $$$$… \\ $$$$\mathrm{answer}\:\mathrm{is}\:\pm\mathrm{24}\sqrt{\mathrm{3}} \\ $$
Commented by infinityaction last updated on 16/Jun/22
$${thank}\:{you}\:{sir} \\ $$
Commented by MJS_new last updated on 16/Jun/22
$$\mathrm{sorry}\:\mathrm{while}\:\mathrm{typing}\:\mathrm{I}\:\mathrm{forgot}\:{a},\:{b},\:{c}\:\mathrm{are}\:\mathrm{positive} \\ $$$$\mathrm{numbers}.\:\Rightarrow\:\mathrm{the}\:\mathrm{only}\:\mathrm{solution}\:\mathrm{is}\:+\mathrm{24}\sqrt{\mathrm{3}} \\ $$