Question Number 130383 by MrHusseinElmasry last updated on 25/Jan/21 $${If}\:\left({a}−\mathrm{2}\right)+\mathrm{3}{i}=\mathrm{5}−{bi}\:{then}\:{a}+{b}= \\ $$ Answered by mathmax by abdo last updated on 25/Jan/21 $$\mathrm{a}−\mathrm{2}+\mathrm{3i}=\mathrm{5}−\mathrm{bi}\:\Rightarrow\mathrm{a}−\mathrm{2}+\mathrm{3i}−\mathrm{5}+\mathrm{bi}=\mathrm{0}\:\Rightarrow\mathrm{a}−\mathrm{7}+\mathrm{i}\left(\mathrm{3}+\mathrm{b}\right)=\mathrm{0}\:\Rightarrow \\ $$$$\begin{cases}{\mathrm{a}−\mathrm{7}=\mathrm{0}}\\{\mathrm{3}+\mathrm{b}=\mathrm{0}\:\:\:\Rightarrow\begin{cases}{\mathrm{a}=\mathrm{7}}\\{\mathrm{b}=−\mathrm{3}\:}\end{cases}}\end{cases} \\…
Question Number 64837 by aliesam last updated on 22/Jul/19 $$\begin{cases}{{x}^{\sqrt{{y}}} \:+\:{y}^{\sqrt{{x}}} \:=\:\frac{\mathrm{49}}{\mathrm{48}}}\\{\sqrt{{x}}\:+\:\sqrt{{y}}\:=\:\frac{\mathrm{7}}{\mathrm{2}}}\end{cases} \\ $$$$ \\ $$$${find}\:{x}\:{and}\:{y} \\ $$$$ \\ $$ Answered by MJS last updated…
Question Number 130362 by greg_ed last updated on 24/Jan/21 $$\mathrm{f}\left({x}\right)=\frac{\mathrm{2}{x}+\mathrm{1}}{\:\sqrt{{x}^{\mathrm{2}} −\mid\mathrm{2}{x}−\mathrm{3}\mid}} \\ $$$$\mathrm{Domain}\:\mathrm{D}_{\mathrm{f}} \:=\:? \\ $$ Answered by ajfour last updated on 24/Jan/21 $${x}^{\mathrm{2}} >\mid\mathrm{2}{x}−\mathrm{3}\mid…
Question Number 130354 by greg_ed last updated on 24/Jan/21 $$\mathrm{f}\left({x}\right)=\frac{\mathrm{4}{x}^{\mathrm{2}} +\mathrm{1}}{\mathrm{2}{x}^{\mathrm{2}} +\mathrm{1}} \\ $$$$\mathrm{prove}\:\mathrm{that}\:\mathrm{1}\:\leqslant\:\mathrm{f}\left({x}\right)\:\leqslant\:\mathrm{2} \\ $$ Answered by MJS_new last updated on 24/Jan/21 $${f}\left({x}\right)=\mathrm{2}−\frac{\mathrm{1}}{\mathrm{2}{x}^{\mathrm{2}} +\mathrm{1}}…
Question Number 130334 by pete last updated on 24/Jan/21 $$\mathrm{Complete}\:\mathrm{the}\:\mathrm{square}\:\mathrm{in}\:\mathrm{the}\:\mathrm{expression} \\ $$$$\mathrm{y}^{\mathrm{2}} \:+\mathrm{8y}+\mathrm{9k}\:\mathrm{and}\:\mathrm{hence}\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$$\mathrm{k}\:\mathrm{that}\:\mathrm{makes}\:\mathrm{it}\:\mathrm{a}\:\mathrm{perfect}\:\mathrm{square}. \\ $$ Answered by TheSupreme last updated on 24/Jan/21 $$\left({y}+{a}\right)={y}^{\mathrm{2}}…
Question Number 130331 by ayoubbacmath0 last updated on 24/Jan/21 $$\mathrm{f}\left(\mathrm{x}\right)={a}\mathrm{x}^{\mathrm{3}} +\mathrm{bx}^{\mathrm{2}} +\mathrm{c} \\ $$$$\mathrm{f}\:'\left(\mathrm{x}\right)=\mathrm{3}{a}\mathrm{x}^{\mathrm{2}} +\mathrm{2bx} \\ $$$$\mathrm{f}\left(\mathrm{0}\right)=−\mathrm{2} \\ $$$$\Rightarrow{a}\left(\mathrm{0}\right)^{\mathrm{3}} +\mathrm{b}\left(\mathrm{0}\right)^{\mathrm{2}} +\mathrm{c}=−\mathrm{2} \\ $$$$\Rightarrow\mathrm{c}=−\mathrm{2} \\ $$$$\mathrm{f}\left(−\mathrm{2}\right)=\mathrm{2}…
Question Number 64758 by Tawa1 last updated on 21/Jul/19 $$\mathrm{Solve}:\:\:\:\:\:\mathrm{x}^{\mathrm{4}} \:+\:\mathrm{5x}^{\mathrm{3}} \:−\:\mathrm{4x}^{\mathrm{2}} \:+\:\mathrm{7x}\:−\:\mathrm{1}\:\:=\:\:\mathrm{0} \\ $$ Answered by ajfour last updated on 21/Jul/19 $${x}={t}−\frac{\mathrm{5}}{\mathrm{4}} \\ $$$${say}\:\:{we}\:{then}\:{get}…
Question Number 130290 by bramlexs22 last updated on 24/Jan/21 $$\:\underset{\mathrm{i}=\mathrm{0}} {\overset{\infty} {\sum}}\underset{\mathrm{j}=\mathrm{0}} {\overset{\infty} {\sum}}\underset{\mathrm{k}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{i}+\mathrm{j}+\mathrm{k}} }\:?\:\mathrm{where}\:\mathrm{i}\neq\mathrm{j}\neq\mathrm{k} \\ $$ Answered by EDWIN88 last updated on…
Question Number 130257 by pete last updated on 23/Jan/21 $${n}\:\mathrm{is}\:\mathrm{an}\:\mathrm{integer} \\ $$$$\mathrm{prove}\:\mathrm{algebraically}\:\mathrm{that}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\: \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}{n}\left({n}+\mathrm{1}\right)\:\mathrm{and}\:\frac{\mathrm{1}}{\mathrm{2}}\left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)\:\mathrm{is}\:\mathrm{always} \\ $$$$\mathrm{a}\:\mathrm{square}\:\mathrm{number}. \\ $$$$\mathrm{note}:\:{write}\:{your}\:{expression}\:{in}\:{a}\:{form} \\ $$$${that}\:{clearly}\:{shows}\:{a}\:{square}\:{number}. \\ $$ Commented by Dwaipayan…
Question Number 130242 by Raxreedoroid last updated on 23/Jan/21 $$\mathrm{Find}\:\:\:{m} \\ $$$${n}=\left({p}_{\mathrm{1}} {p}_{\mathrm{2}} \right)^{{m}} \\ $$$$\mathrm{2}{n}=\left({p}_{\mathrm{1}} ^{\mathrm{1}+{log}_{{p}_{\mathrm{1}} {p}_{\mathrm{2}} } {n}} \right)\left({p}_{\mathrm{2}} ^{\mathrm{1}+{log}_{{p}_{\mathrm{1}} {p}_{\mathrm{2}} } {n}}…