Question Number 43488 by peter frank last updated on 11/Sep/18
$${if}\:{the}\:{root}\:{of}\:\:{x}^{\mathrm{3}} +{px}_{} ^{\mathrm{2}} +{qx}+\mathrm{30}=\mathrm{0}\:{are} \\ $$$${in}\:{the}\:{ratio}\:\mathrm{2}:\mathrm{3}:\mathrm{5}{find}\:{the}\:{value}\:{of}\:{p}\:{and}\:{q} \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 11/Sep/18
$${roots}\:{are}\:\left(\mathrm{2}{k},\mathrm{3}{k}\:{and}\:\mathrm{5}{k}\right) \\ $$$$\mathrm{2}{k}+\mathrm{3}{k}+\mathrm{5}{k}=\frac{−{p}}{\mathrm{1}} \\ $$$$\mathrm{2}{k}.\mathrm{3}{k}.+\mathrm{2}{k}.\mathrm{5}{k}++\mathrm{3}{k}.\mathrm{5}{k}={q} \\ $$$$\mathrm{2}{k}×\mathrm{3}{k}×\mathrm{5}{k}=−\mathrm{30} \\ $$$$\mathrm{30}{k}^{\mathrm{3}} =−\mathrm{30} \\ $$$${k}=−\mathrm{1} \\ $$$$ \\ $$$$\:\mathrm{10}{k}=−{p} \\ $$$${p}=−\mathrm{10}×−\mathrm{1}=\mathrm{10} \\ $$$${q}=\mathrm{31}{k}^{\mathrm{2}} =\mathrm{31}×\left(−\mathrm{1}\right)^{\mathrm{2}} \:.\:=\mathrm{31} \\ $$$$ \\ $$$$ \\ $$
Commented by peter frank last updated on 11/Sep/18
$${thank}\:{you}\:{sir} \\ $$