Question Number 226513 by aba_math last updated on 01/Dec/25 $${Find}\:{gcd}\left({a}^{\mathrm{2}} +{ab}+{b}^{\mathrm{2}} ,{ab}\right)\:{if}\:{gcd}\left({a},{b}\right)=\mathrm{1} \\ $$ Answered by peace2 last updated on 02/Dec/25 $${a}={dx};{b}={dy};{gcd}\left({x},{y}\right)=\mathrm{1} \\ $$$${gcd}\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}}…
Question Number 226455 by Spillover last updated on 29/Nov/25 Answered by Ghisom_ last updated on 29/Nov/25 $$\frac{\mathrm{2}−{x}^{\mathrm{2}} }{\left(\mathrm{1}−{x}\right)\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}=\frac{\mathrm{1}+\mathrm{1}−{x}^{\mathrm{2}} }{\left(\mathrm{1}−{x}\right)\sqrt{\left(\mathrm{1}−{x}\right)\left(\mathrm{1}+{x}\right)}}= \\ $$$$=\frac{\mathrm{1}}{\left(\mathrm{1}−{x}\right)^{\mathrm{3}/\mathrm{2}} \left(\mathrm{1}+{x}\right)^{\mathrm{1}/\mathrm{2}} }+\frac{\left(\mathrm{1}−{x}\right)\left(\mathrm{1}+{x}\right)}{\left(\mathrm{1}−{x}\right)^{\mathrm{3}/\mathrm{2}} \left(\mathrm{1}+{x}\right)^{\mathrm{1}/\mathrm{2}}…
Question Number 226464 by Spillover last updated on 29/Nov/25 Commented by Frix last updated on 29/Nov/25 $$\mathrm{By}\:\mathrm{parts} \\ $$$${u}'=\mathrm{1}\:\rightarrow\:{u}={x} \\ $$$${v}=\mathrm{cot}^{−\mathrm{1}} \:\left({x}^{\mathrm{2}} −{x}+\mathrm{1}\right)\:\rightarrow\:{v}'=−\frac{\mathrm{2}{x}−\mathrm{1}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)\left({x}^{\mathrm{2}} −\mathrm{2}{x}+\mathrm{2}\right)}…
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Question Number 226006 by Linton last updated on 18/Nov/25 $$\left(\mathrm{3}/\mathrm{7}\right)^{\mathrm{0}} \:\:\:{prove}\:{and}\:{evalute}\:{show}\:{all} \\ $$$${working} \\ $$ Commented by mr W last updated on 18/Nov/25 $${for}\:{any}\:{a}\neq\mathrm{0}\:{we}\:{have} \\…
Question Number 225599 by Jubr last updated on 04/Nov/25 Commented by Frix last updated on 04/Nov/25 $$\mathrm{If}\:{a},\:{b},\:{c},\:{d}\:\in\mathbb{R}\:\mathrm{no}\:\mathrm{maximum}\:\mathrm{exists}. \\ $$$$\mathrm{Let}\:{a}={b}=−{r};\:{c}=\mathrm{1};\:{d}=\mathrm{2}{r} \\ $$$$\left(\mathrm{1}−{r}\right)^{\mathrm{2}} \left(\mathrm{1}+\mathrm{2}{r}\right)=\mathrm{1}−\mathrm{3}{r}^{\mathrm{2}} +\mathrm{2}{r}^{\mathrm{3}} \\ $$$$\underset{{r}\rightarrow+\infty}…
Question Number 224691 by lockedin last updated on 27/Sep/25 $$\mathrm{3}{k}+\mathrm{4}={n}^{\mathrm{2}} .\:{k},{n}\:\in\mathbb{N} \\ $$$${Find}\:{all}\:{n}\:{numbers}\:. \\ $$ Answered by Rasheed.Sindhi last updated on 27/Sep/25 $$\mathrm{3}{k}+\mathrm{4}={n}^{\mathrm{2}} .\:{k},{n}\:\in\mathbb{N} \\…
Question Number 224443 by som(math1967) last updated on 12/Sep/25 $$ \\ $$$$\boldsymbol{{S}}{ame}\:{problem}\:{with}\:{me} \\ $$$${please}\:{fix}\:{the}\:{problem} \\ $$ Answered by Tinku Tara last updated on 12/Sep/25 $$\mathrm{The}\:\mathrm{problem}\:\mathrm{is}\:\mathrm{fixed}\:\mathrm{and}\:\mathrm{a}\:\mathrm{new}\:\mathrm{update}…
Question Number 224305 by gregori last updated on 01/Sep/25 Answered by fkwow344 last updated on 01/Sep/25 $$\mathrm{Let}'\mathrm{s}\:\mathrm{set}\:\mathrm{as}\:\overset{\rightarrow} {\boldsymbol{\mathrm{v}}}_{\mathrm{1}} =\left(\mathrm{2},\mathrm{1}\right)^{\intercal} \:,\:\overset{\rightarrow} {\boldsymbol{\mathrm{v}}}_{\mathrm{2}} =\left(\mathrm{1},\mathrm{0}\right)^{\intercal} \\ $$$${A}={PJP}^{−\mathrm{1}} \:\left(\mathrm{Jordan}\:\mathrm{decomposition}\right)…
Question Number 224100 by Jgrads last updated on 19/Aug/25 $$\mathrm{Calculate}\:\mathrm{I}=\underset{\:\mathrm{0}} {\int}^{\:+\infty} \left[\frac{\mathrm{1}}{\mathrm{t}}−\frac{\mathrm{1}}{\mathrm{sh}\left(\mathrm{t}\right)}\right]^{\:\mathrm{2}} \mathrm{dt} \\ $$ Answered by MathematicalUser2357 last updated on 28/Aug/25 $$\int_{\mathrm{0}} ^{\infty} \left(\frac{\mathrm{1}}{{t}}−\frac{\mathrm{1}}{\mathrm{sinh}\:{t}}\right)^{\mathrm{2}}…