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The-equation-x-2-2xp-q-0-and-x-2-2ax-b-0-have-common-roots-show-that-q-b-2-4-a-p-aq-pb-0-

Question Number 159653 by MathsFan last updated on 19/Nov/21 $$\:{The}\:{equation}\:{x}^{\mathrm{2}} +\mathrm{2}{xp}+{q}=\mathrm{0} \\ $$$$\:{and}\:{x}^{\mathrm{2}} +\mathrm{2}{ax}+{b}=\mathrm{0}\:{have}\:{common} \\ $$$${roots},\:{show}\:{that}\:\left({q}−{b}\right)^{\mathrm{2}} +\mathrm{4}\left({a}−{p}\right)\left({aq}−{pb}\right)=\mathrm{0} \\ $$$$ \\ $$ Answered by Rasheed.Sindhi last…

The-result-of-adding-the-odd-natural-numbers-is-1-1-1-3-4-1-3-5-9-1-3-5-7-16-1-3-5-7-9-25-show-that-from-this-result-i-1-n-2i-1-n-2-

Question Number 94096 by Rio Michael last updated on 16/May/20 $$\mathrm{The}\:\mathrm{result}\:\mathrm{of}\:\mathrm{adding}\:\mathrm{the}\:\mathrm{odd}\:\mathrm{natural}\:\mathrm{numbers}\:\mathrm{is}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}\:=\:\mathrm{1} \\ $$$$\:\:\:\:\:\:\:\:\mathrm{1}\:+\:\mathrm{3}=\mathrm{4} \\ $$$$\:\:\mathrm{1}\:+\:\mathrm{3}+\:\mathrm{5}\:=\:\mathrm{9} \\ $$$$\mathrm{1}\:+\:\mathrm{3}\:+\:\mathrm{5}\:+\mathrm{7}\:=\:\mathrm{16} \\ $$$$\mathrm{1}\:+\:\mathrm{3}\:+\:\mathrm{5}\:+\:\mathrm{7}+\mathrm{9}\:=\:\mathrm{25} \\ $$$$\:\mathrm{show}\:\mathrm{that}\:\mathrm{from}\:\mathrm{this}\:\mathrm{result},\:\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\left(\mathrm{2}{i}−\mathrm{1}\right)\:=\:{n}^{\mathrm{2}}…