Question Number 226053 by Linton last updated on 18/Nov/25
$${The}\:{coefficient}\:{of}\:{x}^{\mathrm{2}} \:{in}\:{the}\:{expansion} \\ $$$${of}\:\left(\mathrm{1}+\:\left(\mathrm{2}/{p}\right){x}\right)^{\mathrm{5}} \:+\:\left(\mathrm{1}+{px}\right)^{\mathrm{6}} \:{is}\:\mathrm{70}. \\ $$$${Find}\:{the}\:{possible}\:{values}\:{of}\:{the} \\ $$$${constant}\:{p}. \\ $$
Answered by mr W last updated on 18/Nov/25
$${coef}.\:{of}\:{x}^{\mathrm{2}} \:{is} \\ $$$${C}_{\mathrm{2}} ^{\mathrm{5}} \left(\frac{\mathrm{2}}{{p}}\right)^{\mathrm{2}} +{C}_{\mathrm{2}} ^{\mathrm{6}} {p}^{\mathrm{2}} =\mathrm{70} \\ $$$$\frac{\mathrm{40}}{{p}^{\mathrm{2}} }+\mathrm{15}{p}^{\mathrm{2}} =\mathrm{70} \\ $$$$\mathrm{3}{p}^{\mathrm{4}} −\mathrm{14}{p}^{\mathrm{2}} +\mathrm{8}=\mathrm{0} \\ $$$$\left({p}^{\mathrm{2}} −\mathrm{4}\right)\left(\mathrm{3}{p}^{\mathrm{2}} −\mathrm{2}\right)=\mathrm{0} \\ $$$${p}^{\mathrm{2}} =\mathrm{4},\:\mathrm{2} \\ $$$$\Rightarrow{p}=\pm\mathrm{2},\:\pm\sqrt{\mathrm{2}} \\ $$