eb879d…: “Onto a winter eyes”

$\newenvironment {prompt}{}{} \newcommand {\ungraded }[0]{} \newcommand {\celsius }[0]{} \newcommand {\degree }[0]{} \newcommand {\ohm }[0]{} \newcommand {\log }[0]{\ln } \newcommand {\newrobustcmd }[0]{} \newcommand {\csshow }[1]{\begingroup \expandafter \endgroup \expandafter \show \csname #1\endcsname } \newcommand {\csmeaning }[1]{\ifcsname #1\endcsname \expandafter \meaning \csname #1\endcsname \else \detokenize {undefined}\fi } \newcommand {\ifdefmacro }[0]{} \newcommand {\ifdefparam }[0]{} \newcommand {\ifdefprotected }[0]{} \newcommand {\ifnumequal }[1]{\ifnumcomp {#1}=} \newcommand {\ifnumgreater }[1]{\ifnumcomp {#1}>} \newcommand {\ifnumless }[1]{\ifnumcomp {#1}<} \newcommand {\ifdimequal }[1]{\ifdimcomp {#1}=} \newcommand {\ifdimgreater }[1]{\ifdimcomp {#1}>} \newcommand {\ifdimless }[1]{\ifdimcomp {#1}<} \newcommand {\expandonce }[1]{\unexpanded \expandafter {#1}} \newcommand {\csexpandonce }[1]{\expandafter \expandonce \csname #1\endcsname } \newcommand {\protecting }[0]{} \newcommand {\csdef }[1]{\expandafter \def \csname #1\endcsname } \newcommand {\csedef }[1]{\expandafter \edef \csname #1\endcsname } \newcommand {\csgdef }[1]{\expandafter \gdef \csname #1\endcsname } \newcommand {\csxdef }[1]{\expandafter \xdef \csname #1\endcsname } \newcommand {\cslet }[2]{\expandafter \let \csname #1\endcsname #2} \newcommand {\letcs }[2]{\ifcsdef {#2} {\expandafter \let \expandafter #1\csname #2\endcsname } {\undef #1}} \newcommand {\csletcs }[2]{\ifcsdef {#2} {\expandafter \let \csname #1\expandafter \endcsname \csname #2\endcsname } {\csundef {#1}}} \newcommand {\csuse }[1]{\ifcsname #1\endcsname \csname #1\expandafter \endcsname \fi } \newcommand {\appto }[2]{\ifundef {#1} {\edef #1{\unexpanded {#2}}} {\edef #1{\expandonce #1\unexpanded {#2}}}} \newcommand {\eappto }[2]{\ifundef {#1} {\edef #1{#2}} {\edef #1{\expandonce #1#2}}} \newcommand {\gappto }[2]{\ifundef {#1} {\xdef #1{\unexpanded {#2}}} {\xdef #1{\expandonce #1\unexpanded {#2}}}} \newcommand {\xappto }[2]{\ifundef {#1} {\xdef #1{#2}} {\xdef #1{\expandonce #1#2}}} \newcommand {\preto }[2]{\ifundef {#1} {\edef #1{\unexpanded {#2}}} {\edef #1{\unexpanded {#2}\expandonce #1}}} \newcommand {\epreto }[2]{\ifundef {#1} {\edef #1{#2}} {\edef #1{#2\expandonce #1}}} \newcommand {\gpreto }[2]{\ifundef {#1} {\xdef #1{\unexpanded {#2}}} {\xdef #1{\unexpanded {#2}\expandonce #1}}} \newcommand {\xpreto }[2]{\ifundef {#1} {\xdef #1{#2}} {\xdef #1{#2\expandonce #1}}} \newcommand {\csappto }[1]{\expandafter \appto \csname #1\endcsname } \newcommand {\cseappto }[1]{\expandafter \eappto \csname #1\endcsname } \newcommand {\csgappto }[1]{\expandafter \gappto \csname #1\endcsname } \newcommand {\csxappto }[1]{\expandafter \xappto \csname #1\endcsname } \newcommand {\cspreto }[1]{\expandafter \preto \csname #1\endcsname } \newcommand {\csepreto }[1]{\expandafter \epreto \csname #1\endcsname } \newcommand {\csgpreto }[1]{\expandafter \gpreto \csname #1\endcsname } \newcommand {\csxpreto }[1]{\expandafter \xpreto \csname #1\endcsname } \newcommand {\csnumdef }[1]{\expandafter \numdef \csname #1\endcsname } \newcommand {\csnumgdef }[1]{\expandafter \numgdef \csname #1\endcsname } \newcommand {\csdimdef }[1]{\expandafter \dimdef \csname #1\endcsname } \newcommand {\csdimgdef }[1]{\expandafter \dimgdef \csname #1\endcsname } \newcommand {\csgluedef }[1]{\expandafter \gluedef \csname #1\endcsname } \newcommand {\csgluegdef }[1]{\expandafter \gluegdef \csname #1\endcsname } \newcommand {\mudef }[2]{\ifundef #1{\def #1{0mu}}{}\edef #1{\the \muexpr #2}} \newcommand {\csmudef }[1]{\expandafter \mudef \csname #1\endcsname } \newcommand {\csmugdef }[1]{\expandafter \mugdef \csname #1\endcsname } \newcommand {\listbreak }[0]{} \newcommand {\listadd }[2]{\ifblank {#2}{}{\appto #1{#2|}}} \newcommand {\listgadd }[2]{\ifblank {#2}{}{\gappto #1{#2|}}} \newcommand {\listcsadd }[1]{\expandafter \listadd \csname #1\endcsname } \newcommand {\listcseadd }[1]{\expandafter \listeadd \csname #1\endcsname } \newcommand {\listcsgadd }[1]{\expandafter \listgadd \csname #1\endcsname } \newcommand {\listcsxadd }[1]{\expandafter \listxadd \csname #1\endcsname } \newcommand {\listcsremove }[1]{\expandafter \listremove \csname #1\endcsname } \newcommand {\listcsgremove }[1]{\expandafter \listgremove \csname #1\endcsname } \newcommand {\dolistloop }[0]{\forlistloop \do } \newcommand {\dolistcsloop }[0]{\forlistcsloop \do } \newcommand {\AfterPreamble }[0]{\AtBeginDocument } \newcommand {\mathtoolsset }[1]{\setkeys {\MT_options_name: }{#1}} \newcommand {\refeq }[1]{\textup {\ref {#1}}} \newcommand {\lparen }[0]{(} \newcommand {\rparen }[0]{)} \newcommand {\ordinarycolon }[0]{:} \newcommand {\MT_test_for_tcb_other:nnnnn }[1]{\MH_if:w t#1\relax \expandafter \MH_use_choice_i:nnnn \MH_else: \MH_if:w c#1\relax \expandafter \expandafter \expandafter \MH_use_choice_ii:nnnn \MH_else: \MH_if:w b#1\relax \expandafter \expandafter \expandafter \expandafter \expandafter \expandafter \expandafter \MH_use_choice_iii:nnnn \MH_else: \expandafter \expandafter \expandafter \expandafter \expandafter \expandafter \expandafter \MH_use_choice_iv:nnnn \MH_fi: \MH_fi: \MH_fi: } \newcommand {\newcases }[6]{\newenvironment {#1}{\MT_start_cases:nnnn {#2}{#3}{#4}{#5}}{\MH_end_cases: \right #6}} \newcommand {\renewcases }[6]{\renewenvironment {#1}{\MT_start_cases:nnnn {#2}{#3}{#4}{#5}}{\MH_end_cases: \right #6}} \newcommand {\SwapAboveDisplaySkip }[0]{\noalign {\vskip -\abovedisplayskip \vskip \abovedisplayshortskip }} \newcommand {\vdotswithin }[1]{{\mathmakebox [\widthof {\ensuremath {{}#1{}}}][c]{{\vdots }}}} \newcommand {\MTFlushSpaceBelow }[0]{\\\noalign {\nobreak \vskip -\lineskip \vskip -\l_MT_shortvdotswithinadjustbelow_dim \vskip -\origjot \vskip \jot }} \newcommand {\mathmbox }[0]{\mathpalette \MT_mathmbox:nn } \newcommand {\crampedsubstack }[1]{\crampedsubarray {c}#1\endcrampedsubarray } \newcommand {\prescript }[3]{\mathchoice {\MT_prescript_inner: {#1}{#2}{#3}{\scriptstyle }}{\MT_prescript_inner: {#1}{#2}{#3}{\scriptstyle }}{\MT_prescript_inner: {#1}{#2}{#3}{\scriptscriptstyle }}{\MT_prescript_inner: {#1}{#2}{#3}{\scriptscriptstyle }}} \newcommand {\spreadlines }[1]{\setlength {\jot }{#1}\ignorespaces } \newcommand {\newgathered }[4]{\newenvironment {#1}{\def \MT_gathered_pre: {#2}\def \MT_gathered_post: {#3}\def \MT_gathered_env_end: {#4}\MT_gathered_env }{\endMT_gathered_env }} \newcommand {\renewgathered }[4]{\renewenvironment {#1}{\def \MT_gathered_pre: {#2}\def \MT_gathered_post: {#3}\def \MT_gathered_env_end: {#4}\MT_gathered_env }{\endMT_gathered_env }} \newcommand {\lgathered }[0]{\def \MT_gathered_pre: {}\def \MT_gathered_post: {\hfil }\def \MT_gathered_env_end: {}\MT_gathered_env } \newcommand {\rgathered }[0]{\def \MT_gathered_pre: {\hfil }\def \MT_gathered_post: {}\def \MT_gathered_env_end: {}\MT_gathered_env } \newcommand {\gathered }[0]{\def \MT_gathered_pre: {\hfil }\def \MT_gathered_post: {\hfil }\def \MT_gathered_env_end: {}\MT_gathered_env } \newcommand {\splitfrac }[2]{\genfrac {}{}{0pt}{1}{\textstyle #1\quad \hfill }{\textstyle \hfill \quad \mathstrut #2}} \newcommand {\splitdfrac }[2]{\genfrac {}{}{0pt}{0}{#1\quad \hfill }{\hfill \quad \mathstrut #2}} \newcommand {\tabitem }[0]{\makebox [1em][r]{\textbullet ~}} \newcommand {\letterPlus }[0]{\makebox [0pt][l]{$+$}} \newcommand {\letterMinus }[0]{\makebox [0pt][l]{$-$}} \newcommand {\texttt }[1]{#1} \newcommand {\qrset }[1]{\setkeys {qr}{#1}} \newcommand {\tcolorboxenvironment }[2]{\BeforeBeginEnvironment {#1}{\begin {tcolorbox}[savedelimiter={#1},#2]}\AfterEndEnvironment {#1}{\end {tcolorbox}}} \newcommand {\tcbpkgprefix }[0]{} \newcommand {\qrlink }[1]{Access using the url: \par \url {#1} or you can scan the QR-Code below: \par \begin {center} \qrcode [height=2in]{#1} \end {center} } \newcommand {\desmos }[3]{ \begin {tcolorbox}[ fonttitle=\large ,halign title=center,arc=0pt,left*=24px,right=24px,colback=defaultDesmosBackground,colframe=defaultDesmosBorder,colbacktitle= defaultDesmosTitleBackground,coltext=defaultDesmosText,coltitle=defaultDesmosTitle,title=Desmos Interactive Lesson,]Try the Desmos interactive tool at \url {https://www.desmos.com/calculator/#1} or scan the QR-code below:\\ \begin {center} \qrcode [height=2in]{https://www.desmos.com/calculator/#1} \end {center} \end {tcolorbox} } \newcommand {\youtube }[1]{ \begin {tcolorbox}[ fonttitle=\large ,halign title=center,arc=0pt,left*=24px,right=24px,colback=defaultYouTubeBackground,colframe=defaultYouTubeBorder,colbacktitle= defaultYouTubeTitleBackground,coltext=defaultYouTubeText,coltitle=defaultYouTubeTitle,title=YouTube Hosted Video,]Watch the video via YouTube at \url {https://www.youtube.com/watch?v=#1} or scan the QR-code below:\\ \begin {center} \qrcode [height=2in]{https://www.youtube.com/watch?v=#1} \end {center} \end {tcolorbox} } \newcommand {\javascript }[0]{\suppress \codetemp } \newcommand {\code }[0]{\suppress \codetemp } \newcommand {\python }[0]{\suppress \codetemp } \newcommand {\HyperFirstAtBeginDocument }[0]{\AtBeginDocument } \newcommand {\degree }[0]{\ensuremath {^\circ }} \newcommand {\celsius }[0]{\ifmmode ^\circ \mathrm {C}\else $^\circ $C\fi } \newcommand {\ohm }[0]{\ifmmode \Omega \else $\Omega $\fi } \newcommand {\dblcolon }[0]{\vcentcolon \mathrel {\mkern -.9mu}\vcentcolon } \newcommand {\coloneqq }[0]{\vcentcolon \mathrel {\mkern -1.2mu}=} \newcommand {\Coloneqq }[0]{\dblcolon \mathrel {\mkern -1.2mu}=} \newcommand {\coloneq }[0]{\vcentcolon \mathrel {\mkern -1.2mu}\mathrel {-}} \newcommand {\Coloneq }[0]{\dblcolon \mathrel {\mkern -1.2mu}\mathrel {-}} \newcommand {\eqqcolon }[0]{=\mathrel {\mkern -1.2mu}\vcentcolon } \newcommand {\Eqqcolon }[0]{=\mathrel {\mkern -1.2mu}\dblcolon } \newcommand {\eqcolon }[0]{\mathrel {-}\mathrel {\mkern -1.2mu}\vcentcolon } \newcommand {\Eqcolon }[0]{\mathrel {-}\mathrel {\mkern -1.2mu}\dblcolon } \newcommand {\colonapprox }[0]{\vcentcolon \mathrel {\mkern -1.2mu}\approx } \newcommand {\Colonapprox }[0]{\dblcolon \mathrel {\mkern -1.2mu}\approx } \newcommand {\colonsim }[0]{\vcentcolon \mathrel {\mkern -1.2mu}\sim } \newcommand {\Colonsim }[0]{\dblcolon \mathrel {\mkern -1.2mu}\sim } \newcommand {\nuparrow }[0]{\MH_nuparrow: } \newcommand {\ndownarrow }[0]{\MH_ndownarrow: } \newcommand {\bigtimes }[0]{\MH_csym_bigtimes: }$

How You Can (And Should) Get More Practice!

Below is a few practice problems of various difficulty, but you will need considerably more practice than one each. For that reason you should definitely use the green “Try Another” button in the top right corner at least two or three times to complete additional versions of these questions for more practice. You should keep using that button until doing these problems feels straight forward and easy, and then come back after a week or so of doing other stuff and try again to make sure it is still just as easy for you.

Theoretically Easier Difficulty Problem

You want to find two numbers that multiply to $\sage {p1Mult}$ but add to $\sage {p1Add}$

Factor the following quadratic by factoring it’s coefficients. $p(x) = \sage {p1f4} = \answer [validator=factorCheck]{\sage {p1f3}}$

Theoretically Medium Difficulty Problem

You want to find two numbers that multiply to $\sage {p2Mult}$ but add to $\sage {p2Add}$

Be careful to pay attention to negatives.

Factor the following quadratic by factoring it’s coefficients. $p(x) = \sage {p2f4} = \answer [validator=factorCheck]{\sage {p2f3}}$

Theoretically Harder Difficulty Problem

You want to find two numbers that multiply to $\sage {p3Mult}$ but add to $\sage {p3Add}$

Be careful to pay attention to negatives.

Factor the following quadratic by factoring it’s coefficients. $p(x) = \sage {p3f4} = \answer [validator=factorCheck]{\sage {p3f3}}$

Press...	...to do
left/right arrows	Move cursor
shift+left/right arrows	Select region
ctrl+a	Select all
ctrl+x/c/v	Cut/copy/paste
ctrl+z/y	Undo/redo
ctrl+left/right	Add entry to list or column to matrix
shift+ctrl+left/right	Add copy of current entry/column to to list/matrix
ctrl+up/down	Add row to matrix
shift+ctrl+up/down	Add copy of current row to matrix
ctrl+backspace	Delete current entry in list or column in matrix
ctrl+shift+backspace	Delete current row in matrix

Type...	...to get
norm	$\|\|\blue{[?]}\|\|$
text	$\text{\blue{[?]}}$
sym_name	$\backslash\texttt{\blue{[?]}}$
abs	$\left\|\blue{[?]}\right\|$
sqrt	$\sqrt{\blue{[?]}}$
paren	$\left(\blue{[?]}\right)$
floor	$\lfloor \blue{[?]} \rfloor$
factorial	$\blue{[?]}!$
exp	${\blue{[?]}}^{\blue{[?]}}$
sub	${\blue{[?]}}_{\blue{[?]}}$
frac	$\dfrac{\blue{[?]}}{\blue{[?]}}$
int	$\displaystyle\int{\blue{[?]}}d\blue{[?]}$
defi	$\displaystyle\int_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}d\blue{[?]}$
deriv	$\displaystyle\frac{d}{d\blue{[?]}}\blue{[?]}$
sum	$\displaystyle\sum_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
prod	$\displaystyle\prod_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
root	$\sqrt[\blue{[?]}]{\blue{[?]}}$
vec	$\left\langle \blue{[?]} \right\rangle$
mat	$\left(\begin{matrix} \blue{[?]} \end{matrix}\right)$
*	$\cdot$
infinity	$\infty$
arcsin	$\arcsin\left(\blue{[?]}\right)$
arccos	$\arccos\left(\blue{[?]}\right)$
arctan	$\arctan\left(\blue{[?]}\right)$
sin	$\sin\left(\blue{[?]}\right)$
cos	$\cos\left(\blue{[?]}\right)$
tan	$\tan\left(\blue{[?]}\right)$
sec	$\sec\left(\blue{[?]}\right)$
csc	$\csc\left(\blue{[?]}\right)$
cot	$\cot\left(\blue{[?]}\right)$
log	$\log\left(\blue{[?]}\right)$
ln	$\ln\left(\blue{[?]}\right)$
alpha	$\alpha$
beta	$\beta$
gamma	$\gamma$
delta	$\delta$
epsilon	$\epsilon$
zeta	$\zeta$
eta	$\eta$
theta	$\theta$
iota	$\iota$
kappa	$\kappa$
lambda	$\lambda$
mu	$\mu$
nu	$\nu$
xi	$\xi$
omicron	$\omicron$
pi	$\pi$
rho	$\rho$
sigma	$\sigma$
tau	$\tau$
upsilon	$\upsilon$
phi	$\phi$
chi	$\chi$
psi	$\psi$
omega	$\omega$
Gamma	$\Gamma$
Delta	$\Delta$
Theta	$\Theta$
Lambda	$\Lambda$
Xi	$\Xi$
Pi	$\Pi$
Sigma	$\Sigma$
Phi	$\Phi$
Psi	$\Psi$
Omega	$\Omega$

How You Can (And Should) Get More Practice!

Theoretically Easier Difficulty Problem

Theoretically Medium Difficulty Problem

Theoretically Harder Difficulty Problem

Controls

Symbols

Settings