Completing the square


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1 Completing the square mctycompletingsquare In this unit we consider how quadratic expressions can be written in an equivalent form using the technique known as completing the square. This technique has applications in a number of areas,butwewillseeanexampleofitsuseinsolvingaquadraticequation. In order to master the techniques explained here it is vital that you undertake plenty of practice exercisessothatallthisbecomessecondnature. Tohelpyoutoachievethis,theunitincludes a substantial number of such exercises. Afterreadingthistext,and/orviewingthevideotutorialonthistopic,youshouldbeableto: writeaquadraticexpressionasacompletesquare,plusorminusaconstant solve a quadratic equation by completing the square Contents 1. Introduction. Some simple equations. The basic technique 4. Casesinwhichthecoefficientof x isnot Summary of the process 7 6. Solving a quadratic equation by completing the square c mathcentre 009
2 1. Introduction Inthisunitwelookataprocesscalledcompletingthesquare. Itcanbeusedtowritea quadraticexpressioninanalternativeform. Laterintheunitwewillseehowitcanbeusedto solve a quadratic equation.. Some simple equations Considerthequadraticequation x = 9. Wecansolvethisbytakingthesquarerootofboth sides: x = or remembering that when we take the square root there will be two possible answers, one positive andonenegative.thisisoftenwritteninthebrieferform x = ±. Thisprocessforsolving x = 9isverystraightforward,particularlybecause: 9isa squarenumber,or completesquare. Thismeansthatitistheresultofsquaring anothernumber,orterm,inthiscasetheresultofsquaringor. x isacompletesquareitistheresultofsquaring x. So simply squarerooting both sides solves the problem. Considertheequation x = 5. Again,wecansolvethisbytakingthesquarerootofbothsides: x = 5 or 5 Inthisexample,therighthandsideof x = 5,isnotasquarenumber. Butwecanstillsolve theequationinthesameway.itisusuallybettertoleaveyouranswerinthisexactform,rather than use a calculator to give a decimal approximation. Suppose we wish to solve the equation x 7) = Again,wecansolvethisbytakingthesquarerootofbothsides.Thelefthandsideisacomplete squarebecauseitresultsfromsquaring x 7. x 7 = or Byadding7toeachsidewecanobtainthevaluesfor x: x = 7 + or 7 Wecouldwritethisinthebrieferform x = 7 ±. c mathcentre 009
3 Supposewewishtosolve x + ) = 5 Againthelefthandsideisacompletesquare.Takingthesquarerootofbothsides: x + = 5 or 5 Bysubtractingfromeachsidewecanobtainthevaluesfor x: x = + 5 or 5 Exercises 1. Solve the following quadratic equations a) x = 5 b) x = 10 c) x = d) x + 1) = 9 e) x + ) = 16 f) x ) = 100 g) x 1) = 5 h) x + 4) =. The basic technique Nowsupposewewantedtotrytoapplythemethodusedinthethreepreviousexamplesto x + 6x = 4 Ineachofthepreviousexamples,thelefthandsidewasacompletesquare.Thismeansthatin eachcaseittooktheform x + a) or x a).thisisnotthecasenowandsowecannotjust takethesquareroot. Whatwetrytodoinsteadisrewritetheexpressionsothatitbecomesa complete square hence the name completing the square. Observethatcompletesquaressuchas x + a) or x a) canbeexpandedasfollows: Key Point complete squares: x + a) = x + a)x + a) = x + ax + a x a) = x a)x a) = x ax + a c mathcentre 009
4 Wewillusetheseexpansionstohelpustocompletethesquareinthefollowingexamples. Consider the quadratic expression We compare this with the complete square x + 6x 4 x + ax + a Clearlythecoefficientsof x inbothexpressionsarethesame. Wewouldliketomatchuptheterm axwiththeterm 6x.Todothisnotethat amustbe 6, sothat a =. Recall that x + a) = x + ax + a Thenwith a = x + ) = x + 6x + 9 Thismeansthatwhentryingtocompletethesquarefor x +6x 4wecanreplacethefirsttwo terms, x + 6x,by x + ) 9.So x + 6x 4 = x + ) 9 4 = x + ) 1 Wehavenowwrittentheexpression x + 6x 4asacompletesquareplusorminusaconstant. Wehavecompletedthesquare.Itisimportanttonotethattheconstantterm,,inbrackets is half the coefficient of x in the original expression. Supposewewishtocompletethesquareforthequadraticexpression x 8x + 7. Wewanttotrytorewritethissothatittakestheformofacompletesquareplusorminusa constant. We compare x 8x + 7 withthestandardform x ax + a Thecoefficientsof x arethesame.tomakethecoefficientsof xthesamewemustchoose a tobe4.recallthat x a) = x ax + a Thenwith a = 4 x 4) = x 8x + 16 Thismeansthatwhentryingtocompletethesquarefor x 8x+7wecanreplacethefirsttwo terms, x 8x,by x 4) 16. So x 8x + 7 = x 4) = x 4) 9 Wehavenowwrittentheexpression x 8x + 7asacompletesquareplusorminusaconstant. Wehavecompletedthesquare. Againnotethattheconstantterm, 4,inbracketsishalfthe coefficient of x in the original expression. 4 c mathcentre 009
5 Supposewewishtocompletethesquareforthequadraticexpression x + 5x +. Thismeanswewanttotrytorewriteitsothatithastheformofacompletesquareplusor minusaconstant. Intheexampleswehavejustworkedthroughwehaveseenhowthiscanbe donebycomparingwiththestandardforms x + a) and x a). Wewouldliketobeable to complete the square without writing down all the working we did in the previous examples. Thekeypointtorememberisthatthenumberinthebracketofthecompletesquareishalfthe coefficient of x in the quadratic expression. Sowith x + 5x + weknowthatthecompletesquarewillbe x + 5. Thishasthesame ) x and xtermsasthegivenquadraticexpressionbuttheconstanttermisdifferent. Wemust balance the constant term by a) subtracting the extra constant that our complete square has ) 5 introduced, that is,andb)puttingintheconstanttermfromourquadratic,thatis. Putting this together we have x + 5x + = x + 5 ) Tofinishoffwejustcombinethetwoconstants ) 5 + = = 1 4 andso x + 5x + = x + 5 ) 1 4 ) 5 + 1) Wehavenowwrittentheexpression x + 5x + asacompletesquareplusorminusaconstant. Wehavecompletedthesquare. Againnotethattheconstantterm, 5,inbracketsishalfthe coefficient of x in the original expression. The explanation given above is really just an outline of our thought process; when we complete thesquareinpracticewewouldnotwriteitalldown.wewouldprobablygostraighttoequation 1).Theabilitytodothiswillcomewithpractice. Exercises Completing the square for the following quadratic expressions a) x + x + b) x + x + 5 c) x + x 1 d) x + 6x + 8 e) x 6x + 8 f) x + x + 1 g) x x 1 h) x + 10x 1 i) x + 5x + 4 j) x + 6x + 9 k) x x + 6 l) x x Cases in which the coefficient of x is not 1. We now know how to complete the square for quadratic expressions for which the coefficient of x is1.whenfacedwithaquadraticexpressionwherethecoefficientof x isnot1wecanstill usethistechniquebutweputinanextrastepfirstwefactoroutthiscoefficient. 5 c mathcentre 009
6 Supposewewishtocompletethesquarefortheexpression x 9x Webeginbyfactoringoutthecoefficientof x,inthiscase.itdoesnotmatterthatisnot afactorof50;wecanstilldothisbywritingtheexpressionas x x + 50 ) Nowtheexpressioninbracketsisaquadraticwithcoefficientof x equalto1andsowecan proceedasbefore.thenumberinthecompletesquarewillbehalfthecoefficientof x,sowewill use x.thenwemustbalanceuptheconstanttermjustaswedidbeforebysubtracting ) ) the extra constant we have introduced, that is,andputtingintheconstantfromthe quadraticexpression,thatis 50. { x x + 50 } = { x ) Thearithmetictotidyuptheconstantsisabitmessy: ) } + 50 So putting all this together ) + 50 = = = 17 1 x x + 50 = x ) and finally and we have completed the square. x x + 50 ) = x ) ) This is the completing the square form for a quadratic expression for which the coefficient of x isnot1. Exercises Completing the square for the following quadratic expressions a) x + 4x 8 b) 5x + 10x + 15 c) x 7x + 9 d) x + 6x + 1 e) x 1x + f) 15 10x x g) 4 + 1x x h) 9 + 6x x 6 c mathcentre 009
7 5. Summary of the process Itwillbeusefulifyoucangetusedtodoingthisprocessautomatically. Themethodcanbe summarised as follows: Key Point 1.factoroutthecoefficientof x thenworkwiththequadraticexpressionwhichhasa coefficientof x equalto1.checkthecoefficientof xinthenewquadraticexpressionandtakehalfofitthisisthe number that goes into the complete square bracket.balancetheconstanttermbysubtractingthesquareofthenumberfromstep,and putting in the constant from the quadratic expression 4. the rest is arithmetic that may often involve fractions 6. Solving a quadratic equation by completing the square Letusreturnnowtoaproblemposedearlier.Wewanttosolvetheequation x + 6x = 4. Wewritethisas x + 6x 4 = 0.Notethatthecoefficientof x is1sothereisnoneedtotake out any common factor. Completing the square for quadratic expression on the lefthand side: x + 6x 4 = 0 x + ) 9 4 = 0 1) x + ) 1 = 0 ) x + ) = 1 x + = ± 1 x = ± 1 We have solved the quadratic equation by completing the square. Toproduceequation1)wehavenotedthatthecoefficientof xinthequadraticexpressionis6 sothenumberinthe completesquare bracketmustbe;thenwehavebalancedtheconstant bysubtractingthesquareofthisnumber,,andputtingintheconstantfromthequadratic, 4. To get equation) we just do the arithmetic which in this example is quite straightforward. 7 c mathcentre 009
8 Exercises 4 Use completing the square to solve the following quadratic equations Answers a) x + 4x 1 = 0 b) x + 5x 6 = 0 c) 10x + 7x 1 = 0 d) x + 4x 8 = 0 e) x x = 0 f) x + 8x 5 = 0 Giveyouranswerseitherasfractionsorintheform p ± q 1. a) ±5 b) ± 10 c) ± d),4 e) 1,7 f) 1,8 g) 1 ± 5 h) 4 ±. a) x + 1) + 1 b) x + 1) + 4 c) x + 1) d) x + ) 1 e) x ) 1 f) x + 1 ) + g) x 1 ) 5 h) x + 5) i) x + 5 ) 9 4 j) x + ) k) x 1) + 5 l) a) [ x + 1) 5 ] b) 5 [ x + 1) + ] c) [ d) x + ) ] 7 [ e) x ) 10 ] 4 g) [ x ) 1 ] h) [ x 1) 4 ] [ x 9 x ) ] 69 4 f) [ x + 5) 40 ] ) a), 6 b) 1, 6 c), 4 5 d) ± 1 e) ± 19 f) ± 8 c mathcentre 009
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