Question Number 107385 by bemath last updated on 10/Aug/20
$$\:\:\:\:\:\iddots\mathcal{B}{e}\mathcal{M}{ath}\iddots \\ $$ $${Given}\:\mathrm{6}{x}^{\mathrm{2}} −\mathrm{6}{px}+\mathrm{14}{p}−\mathrm{2}=\mathrm{0} \\ $$ $${has}\:{the}\:{roots}\:{are}\:\:{u}\:\&\:{v}\:{where}\:{u},{v}\:\notin\mathbb{Z} \\ $$ $${If}\:{u},{v}\:\geqslant\:\mathrm{1}\:,\:{then}\:{the}\:{value}\:{of}\:\mid{u}−{v}\mid\:. \\ $$ $$\left({a}\right)\mathrm{14}\:\:\:\:\:\left({b}\right)\mathrm{15}\:\:\:\:\:\left({c}\right)\mathrm{16}\:\:\:\:\:\left({d}\right)\mathrm{17}\:\:\:\left({e}\right)\:\mathrm{18} \\ $$
Commented byHer_Majesty last updated on 10/Aug/20
$${u}=\frac{{p}}{\mathrm{2}}−\frac{\sqrt{\mathrm{9}{p}^{\mathrm{2}} −\mathrm{84}{p}+\mathrm{12}}}{\mathrm{6}} \\ $$ $${v}=\frac{{p}}{\mathrm{2}}+\frac{\sqrt{\mathrm{9}{p}^{\mathrm{2}} −\mathrm{84}{p}+\mathrm{12}}}{\mathrm{6}} \\ $$ $${u}\geqslant\mathrm{1}\wedge{v}\geqslant\mathrm{1}\:\Rightarrow\:{p}\geqslant\frac{\mathrm{14}+\mathrm{2}\sqrt{\mathrm{46}}}{\mathrm{3}} \\ $$ $$\mid{u}−{v}\mid=\frac{\sqrt{\mathrm{9}{p}^{\mathrm{2}} −\mathrm{84}{p}+\mathrm{12}}}{\mathrm{3}} \\ $$ $${and}\:{we}\:{can}\:{find}\:{a}\:{p}\:{for}\:{any}\:{given}\:{value}\:{of}\:{A} \\ $$ $${with} \\ $$ $${A}=\frac{\sqrt{\mathrm{9}{p}^{\mathrm{2}} −\mathrm{84}{p}+\mathrm{12}}}{\mathrm{3}}\:\Leftrightarrow\:{p}=\frac{\mathrm{14}+\sqrt{\mathrm{9}{A}^{\mathrm{2}} +\mathrm{184}}}{\mathrm{3}} \\ $$ $${and}\:{p}\geqslant\frac{\mathrm{14}+\mathrm{2}\sqrt{\mathrm{46}}}{\mathrm{3}} \\ $$ $${so}\:{either}\:{the}\:{question}\:{is}\:{strange}\:{or}\:{I}\:{am} \\ $$ $${wrong}\:\left({where}?\right) \\ $$
Commented bybemath last updated on 10/Aug/20
Commented bybemath last updated on 10/Aug/20
$${the}\:{original}\:{question} \\ $$
Commented bybemath last updated on 10/Aug/20
$${if}\:{u}+{v}\:\geqslant\:\mathrm{2}\:\Rightarrow\:{p}\:\geqslant\:\mathrm{2} \\ $$ $${u}×{v}\:\geqslant\:\mathrm{1}\:\Rightarrow\frac{\mathrm{14}{p}−\mathrm{2}}{\mathrm{6}}\:\geqslant\:\mathrm{1}\:\Rightarrow{p}\:\geqslant\:\frac{\mathrm{4}}{\mathrm{7}} \\ $$ $$\left(\mathrm{1}\right)\wedge\left(\mathrm{2}\right)\:\Rightarrow\:{p}\:\geqslant\:\mathrm{2} \\ $$
Commented bybemath last updated on 10/Aug/20
$${how}\:{we}\:{get}\:{the}\:{value}\:{of}\:{u}\:{and}\:{v}? \\ $$
Commented bybemath last updated on 10/Aug/20
$${the}\:{roots}\:{are}\:{not}\:{integer} \\ $$
Commented byhgrocks last updated on 10/Aug/20
