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7/31/2019 ETAD 803 Assignment 1 - Completing the Square
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Comple'ngthesquareisaprocessthatisusuallycoveredingrade11mathcourses.
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Thislessonwillfocusonlyonquadra'cswithaleadingcoefficientora-valueof1. Forexample,y=x2+8x-7andnoty=2x2+8x7. Thereisanimportantextrastepinthatprocedurethatisrequiredtohave
adifferentleadingcoefficient,anditwillnotbecoveredinthisbasic
introduc'on
Anexampleofaquadra'cinvertexformisy=(x+3)2+5. Studentswillhavepreviousexperiencegraphingquadra'csinvertexform. Emphasizethebenefitofhavingthequadra'cinvertexformtoallowforeasier
graphing.
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Thesestepsareveryspecificandshouldbecompletedinthisorder.
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Thisstepissome'mesalreadycompletedforyou.Inthosecases,youdonotneedtodoanythinginstep1.
Besurethatyoury-valueisposi've,ifitisnot,thendividealltermsbynega've1orchangeallofthesignsfrom+to-orviceversa.
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Simplyplacebracketsaroundthefirsttwoterms. Notthatinthecasewearecoveringhereawillalwaysbeequalto1andwillnot
bewriPenintheques'on.
brepresentsanumericalcoefficient.
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Besuretodothisworktothesideoftheques'onastonotmuddleupthestringofchangestotheoriginalfunc'on.
Notethatthisvaluewillalwaysendupposi'vebecauseyouaresquaringanumber.
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Inthisstepweareaddingthisvalueinordertoturntheexpressionwithinthebracketsintoaperfecttrinomialsquare.
Thisiswhytheprocessiscalledcomple'ngthesquare.
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ThisstepiseasilyforgoPen. Stresstheimportanceofthisstepbymakingitveryclearthattheequa'on
cannotbechangedandmustremainbalanced.
Byaddingandthensubtrac'ngthesamevalue,wearenotchangingtheequa'onbecauseweareaddingzero(e.g.+4-4=0).
Addingzerodoesnotchangetheequa'on. Ifweforgettosubtractthisvalue,thenweareinfactchangingthe
equa'on.
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Inthisstep,weareusingourpreviousknowledgeoffactoringtofactortheperfecttrinomialsquarewithinthebrackets.
Ifthepreviousstepshavebeencompletedcorrectly,thenthefactoringshouldalwaysworkouttobethesamebinomialtwice.
Anothertricktothisfactoringisthatthenumberisalwaysgoingtobeb2. Donotteachthisdirectly,butratherallowthestudentstodiscoverthe
paPernontheirown.
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Inthisstepyoujustaddthetermsoutsidethebrackettogether. Rememberthatthefirsttermisanega've.
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Thisstepisheretoremindstudentsofthereasonforcomple'ngthesquare. Vertexformisverygoodforgraphing.
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Thisisjustasummaryofthestepswithadifferentexampleshown.
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Thisisjustasummaryofthestepswithadifferentexampleshown.
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Concludebyremindingstudentswhywewanttobeabletocompletethesquare:itistomakegraphingeasy!
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