Completing the square mc-ty-completingsquare-009-1 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 isnot1 5 5. Summary of the process 7 6. Solving a quadratic equation by completing the square 7 www.mathcentre.ac.uk 1 c mathcentre 009
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 isacompletesquare-itistheresultofsquaring x. So simply square-rooting both sides solves the problem. Considertheequation x = 5. Again,wecansolvethisbytakingthesquarerootofbothsides: x = 5 or 5 Inthisexample,theright-handsideof 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.Theleft-handsideisacomplete squarebecauseitresultsfromsquaring x 7. x 7 = or Byadding7toeachsidewecanobtainthevaluesfor x: x = 7 + or 7 Wecouldwritethisinthebrieferform x = 7 ±. www.mathcentre.ac.uk c mathcentre 009
Supposewewishtosolve x + ) = 5 Againtheleft-handsideisacompletesquare.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,theleft-handsidewasacompletesquare.Thismeansthatin eachcaseittooktheform x + a) or x a).thisisnotthecasenowandsowecannotjust takethesquare-root. 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 www.mathcentre.ac.uk c mathcentre 009
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) 16 + 7 = x 4) 9 Wehavenowwrittentheexpression x 8x + 7asacompletesquareplusorminusaconstant. Wehavecompletedthesquare. Againnotethattheconstantterm, 4,inbracketsishalfthe coefficient of x in the original expression. www.mathcentre.ac.uk 4 c mathcentre 009
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 + = 5 4 + 1 4 = 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 + 1 4. 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 usethistechniquebutweputinanextrastepfirst-wefactoroutthiscoefficient. www.mathcentre.ac.uk 5 c mathcentre 009
Supposewewishtocompletethesquarefortheexpression x 9x + 50. 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 = 9 4 + 50 = 7 1 + 00 1 = 17 1 x x + 50 = x ) + 17 1 and finally and we have completed the square. x x + 50 ) = x ) ) + 17 1 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 www.mathcentre.ac.uk 6 c mathcentre 009
5. Summary of the process Itwillbeusefulifyoucangetusedtodoingthisprocessautomatically. Themethodcanbe summarised as follows: Key Point 1.factoroutthecoefficientof x -thenworkwiththequadraticexpressionwhichhasa coefficientof x equalto1.checkthecoefficientof xinthenewquadraticexpressionandtakehalfofit-thisisthe 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 left-hand 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. www.mathcentre.ac.uk 7 c mathcentre 009
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) 10 + 6x 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) 6 4 4. 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 ] ) 5 4 4. a), 6 b) 1, 6 c), 4 5 d) ± 1 e) ± 19 f) ± www.mathcentre.ac.uk 8 c mathcentre 009