Completing the Square - Math is Fun

Steps · Step 1 Divide all terms by a (the coefficient of x2). · Step 2 Move the number term (c/a) to the right side of the equation. · Step 3 Complete the square ... CompletingtheSquare "CompletingtheSquare"iswherewe... ...takeaQuadraticEquation likethis: andturnit intothis: ax2+bx+c=0 a(x+d)2+e=0 Forthoseofyouinahurry,Icantellyouthat:d=b2a and:e=c−b24a Butifyouhavetime,letmeshowyouhowto"CompletetheSquare"yourself. CompletingtheSquare Saywehaveasimpleexpressionlikex2+bx.Havingxtwiceinthesameexpressioncanmakelifehard.Whatcanwedo? Well,withalittleinspirationfromGeometrywecanconvertit,likethis: Asyoucanseex2+bxcanberearrangednearlyintoasquare... ...andwecancompletethesquarewith(b/2)2 InAlgebraitlookslikethis: x2+bx +(b/2)2 = (x+b/2)2   "Completethe Square"     So,byadding(b/2)2wecancompletethesquare. Theresultof(x+b/2)2hasxonlyonce,whichiseasiertouse. KeepingtheBalance Now...wecan'tjustadd(b/2)2withoutalsosubtractingittoo!Otherwisethewholevaluechanges. Solet'sseehowtodoitproperlywithanexample: Startwith:    ("b"is6inthiscase)     CompletetheSquare:   Alsosubtractthenewterm Simplifyitandwearedone.   Theresult: x2+6x+7 = (x+3)2−2 Andnowxonlyappearsonce,andourjobisdone! AShortcutApproach Hereisaquickwaytogetananswer.Youmaylikethismethod. Firstthinkabouttheresultwewant:(x+d)2+e Afterexpanding(x+d)2weget:x2+2dx+d2+e Nowseeifwecanturnourexampleintothatformtodiscoverdande Example:trytofitx2+6x+7intox2+2dx+d2+e Nowwecan"force"ananswer: Weknowthat6xmustendupas2dx,sodmustbe3 Nextweseethat7mustbecomed2+e=9+e,soemustbe−2 Andwegetthesameresult(x+3)2−2asabove!   Now,letuslookatausefulapplication:solvingQuadraticEquations... SolvingGeneralQuadraticEquationsbyCompletingtheSquare WecancompletethesquaretosolveaQuadraticEquation(findwhereitisequaltozero). ButageneralQuadraticEquationcanhaveacoefficientofainfrontofx2: ax2+bx+c=0 Butthatiseasytodealwith...justdividethewholeequationby"a"first,thencarryon: x2+(b/a)x+c/a=0 Steps NowwecansolveaQuadraticEquationin5steps: Step1Dividealltermsbya(thecoefficientofx2). Step2Movethenumberterm(c/a)totherightsideoftheequation. Step3Completethesquareontheleftsideoftheequationandbalancethisbyaddingthesamevaluetotherightsideoftheequation. Wenowhavesomethingthatlookslike(x+p)2=q,whichcanbesolvedrathereasily: Step4Takethesquarerootonbothsidesoftheequation. Step5Subtractthenumberthatremainsontheleftsideoftheequationtofindx. Examples OK,someexampleswillhelp! Example1:Solvex2+4x+1=0 Step1canbeskippedinthisexamplesincethecoefficientofx2is1 Step2Movethenumbertermtotherightsideoftheequation: x2+4x=-1 Step3Completethesquareontheleftsideoftheequationandbalancethisbyaddingthesamenumbertotherightsideoftheequation. (b/2)2=(4/2)2=22=4 x2+4x+4=-1+4 (x+2)2=3 Step4Takethesquarerootonbothsidesoftheequation: x+2=±√3=±1.73(to2decimals) Step5Subtract2frombothsides: x=±1.73–2=-3.73or-0.27 Andhereisaninterestingandusefulthing. Attheendofstep3wehadtheequation: (x+2)2=3 Itgivesusthevertex(turningpoint)ofx2+4x+1:(-2,-3)     Example2:Solve5x2–4x–2=0 Step1Dividealltermsby5 x2–0.8x–0.4=0 Step2Movethenumbertermtotherightsideoftheequation: x2–0.8x=0.4 Step3Completethesquareontheleftsideoftheequationandbalancethisbyaddingthesamenumbertotherightsideoftheequation: (b/2)2=(0.8/2)2=0.42=0.16 x2–0.8x+0.16=0.4+0.16 (x–0.4)2=0.56 Step4Takethesquarerootonbothsidesoftheequation: x–0.4=±√0.56=±0.748(to3decimals) Step5Subtract(-0.4)frombothsides(inotherwords,add0.4): x=±0.748+0.4=-0.348or1.148 Why"CompletetheSquare"? WhycompletethesquarewhenwecanjustusetheQuadraticFormulatosolveaQuadraticEquation? Well,onereasonisgivenabove,wherethenewformnotonlyshowsusthevertex,butmakesiteasiertosolve. Therearealsotimeswhentheformax2+bx+cmaybepartofalargerquestionandrearrangingitasa(x+d)2+emakesthesolutioneasier,becausexonlyappearsonce. Forexample"x"mayitselfbeafunction(likecos(z))andrearrangingitmayopenupapathtoabettersolution. AlsoCompletingtheSquareisthefirststepintheDerivationoftheQuadraticFormula Justthinkofitasanothertoolinyourmathematicstoolbox.   364,1205,365,2331,2332,3213,3896,3211,3212,1206   Footnote:Valuesof"d"and"e" HowdidIgetthevaluesofdandefromthetopofthepage? Startwith Dividetheequationbya Putc/aonotherside Add(b/2a)2tobothsides   "CompletetheSquare" Nowbringeverythingback... ...totheleftside ...totheoriginalmultipleaofx2 Andyouwillnoticethatwehave: a(x+d)2+e=0 Where:d= b 2a and:e=c− b2 4a Justlikeatthetopofthepage!   QuadraticEquations FactoringQuadratics GraphingQuadraticEquations RealWorldExamplesofQuadraticEquations DerivationofQuadraticEquation QuadraticEquationSolver AlgebraIndex Copyright©2021MathsIsFun.com