Question Number 228265 by ajfour last updated on 04/Apr/26
$$\:\:{Quartic}:\:\:{x}^{\mathrm{4}} +{ax}^{\mathrm{3}} +{bx}^{\mathrm{2}} +{cx}+{d}=\mathrm{0} \\ $$$$\left({x}^{\mathrm{2}} +{px}+{h}\right)\left({x}^{\mathrm{2}} +{qx}+{k}\right)=\mathrm{0} \\ $$$${p}+{q}={a} \\ $$$${h}+{k}={b}−{pq}={b}−{m}\:\:\:\left({pq}={m}\:{say}\right) \\ $$$${pk}+{qh}={c} \\ $$$${hk}={d} \\ $$$${q}\left({h}+{k}\right)={q}\left({b}−{m}\right) \\ $$$${p}\left({h}+{k}\right)={p}\left({b}−{m}\right) \\ $$$${ph}+{qk}={c} \\ $$$${k}\left({p}−{q}\right)={p}\left({b}−{m}\right)−{c} \\ $$$${h}\left({p}−{q}\right)={c}−{q}\left({b}−{m}\right) \\ $$$${multiplying} \\ $$$${d}\left({a}^{\mathrm{2}} −\mathrm{4}{m}\right)= \\ $$$$\:\:\:\:\:\:\:\:\:{ac}\left({b}−{m}\right)−{m}\left({b}−{m}\right)^{\mathrm{2}} −{c}^{\mathrm{2}} \\ $$$${m}\left({b}−{m}\right)^{\mathrm{2}} +\left({ac}−\mathrm{4}{d}\right){m} \\ $$$$\:\:\:\:\:\:\:+{a}\left({ad}−{bc}\right)+{c}^{\mathrm{2}} =\mathrm{0} \\ $$$$−−−−−−−−−−−−−−− \\ $$$$\boldsymbol{{m}}^{\mathrm{3}} −\mathrm{2}\boldsymbol{{bm}}^{\mathrm{2}} +\left(\boldsymbol{{b}}^{\mathrm{2}} +\boldsymbol{{ac}}−\mathrm{4}\boldsymbol{{d}}\right)\boldsymbol{{m}} \\ $$$$\:\:\:\:+\boldsymbol{{a}}\left(\boldsymbol{{ad}}−\boldsymbol{{bc}}\right)+\boldsymbol{{c}}^{\mathrm{2}} =\mathrm{0} \\ $$$$−−−−−−−−−−−−−−− \\ $$$$\boldsymbol{{p}}\:,\:\boldsymbol{{q}}=\frac{\boldsymbol{{a}}}{\mathrm{2}}\pm\sqrt{\frac{\boldsymbol{{a}}^{\mathrm{2}} }{\mathrm{4}}−\boldsymbol{{m}}} \\ $$$$−−−−−−−−−−−−−−− \\ $$$$\boldsymbol{{h}}=\frac{\boldsymbol{{c}}−\boldsymbol{{q}}\left(\boldsymbol{{b}}−\boldsymbol{{m}}\right)}{\boldsymbol{{p}}−\boldsymbol{{q}}} \\ $$$$\boldsymbol{{k}}=\frac{\boldsymbol{{p}}\left(\boldsymbol{{b}}−\boldsymbol{{m}}\right)−\boldsymbol{{c}}}{\boldsymbol{{p}}−\boldsymbol{{q}}} \\ $$$$\:−−−−−−−−−−−−−−− \\ $$$$\boldsymbol{{x}}=−\frac{\boldsymbol{{p}}}{\mathrm{2}}\pm\sqrt{\frac{\boldsymbol{{p}}^{\mathrm{2}} }{\mathrm{4}}−\left\{\frac{\boldsymbol{{c}}−\boldsymbol{{q}}\left(\boldsymbol{{b}}−\boldsymbol{{m}}\right)}{\boldsymbol{{p}}−\boldsymbol{{q}}}\right\}} \\ $$$$\boldsymbol{{or}}\:−−−−−−−−−−−−−− \\ $$$$\boldsymbol{{x}}=−\frac{\boldsymbol{{q}}}{\mathrm{2}}\pm\sqrt{\frac{\boldsymbol{{q}}^{\mathrm{2}} }{\mathrm{4}}−\left\{\frac{\boldsymbol{{p}}\left(\boldsymbol{{b}}−\boldsymbol{{m}}\right)−\boldsymbol{{c}}}{\boldsymbol{{p}}−\boldsymbol{{q}}}\right\}} \\ $$$$−−−−−−−−−−−−−−− \\ $$
Commented by ajfour last updated on 04/Apr/26
https://youtu.be/WMn-9Dg2WiQ?si=VE6YX5HH3aJlKluB
Commented by ajfour last updated on 04/Apr/26
$${I}\:{made}\:{a}\:{video}\:{too}\:{for}\:{this}. \\ $$
Commented by TonyCWX last updated on 05/Apr/26
$$\mathrm{That}'\mathrm{s}\:\mathrm{Wonderful},\:\mathrm{Sir}!! \\ $$