{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 259 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 260 "Times " 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 261 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Blue Emp hasis" -1 262 "Times" 1 12 0 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 261 263 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 261 264 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 261 265 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times " 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 17 "Solving equations" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 23.3.2007 " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 16 "Linear equations" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 20 "The Maple procedure " }{TEXT 0 5 "so lve" }{TEXT -1 46 " can be used to solve many types of equations." }} {PARA 0 "" 0 "" {TEXT -1 52 "For example, we can solve a linear equati on such as " }{XPPEDIT 18 0 "2*x+5 = x/2-7;" "6#/,&*&\"\"#\"\"\"%\"xGF 'F'\"\"&F',&*&F(F'F&!\"\"F'\"\"(F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "solve(2*x + \+ 5 = x/2 - 7);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "To check the solution we can isolate the left and right s ides of the equation using the command " }{TEXT 0 3 "lhs" }{TEXT -1 31 " to get the left-hand side and " }{TEXT 0 3 "rhs" }{TEXT -1 28 " t o get the right-hand side." }}{PARA 0 "" 0 "" {TEXT -1 81 "Note that t he whole equation can be assigned as a data structure to the variable \+ " }{TEXT 0 2 "eq" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "eq := 2*x + 5 = x/2 - 7;\nlh s(eq);\nrhs(eq);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 0 4 "subs" }{TEXT -1 107 " command can be use d to check that the numerical values of the left and right sides when \+ are the same when " }{XPPEDIT 18 0 "x = -8;" "6#/%\"xG,$\"\")!\"\"" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "subs(x=-8,lhs(eq));\nsubs(x=-8,rhs(eq));" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "Alternati vely just substitute x=-8 in the whole equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "subs(x=-8,eq );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "We end up with a true statement." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 99 "We can mechanically test whether a statem ent or relation such as this is true by using the command " }{TEXT 0 5 "evalb" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 15 "The Maple wor d " }{TEXT 0 5 "evalb" }{TEXT -1 35 " can be thought of as standing fo r " }{TEXT 259 41 "evaluate as a Boolean or logical variable" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "subs(x=-8,eq);\nevalb(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 117 "Since the equation only has th e one solution x = -8, substituting any other value of x will produce \+ a false relation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "eq := 2*x + 5 = x/2 - 7;\nsubs(x=2,eq);\neval b(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "For another example consider the equation " }{XPPEDIT 18 0 "(t-3)/4 \+ = 5*t-7;" "6#/*&,&%\"tG\"\"\"\"\"$!\"\"F'\"\"%F),&*&\"\"&F'F&F'F'\"\"( F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 127 "The following comm ands solve this equation, assign the solution to the variable t2 and t hen check the solution by substitution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "eq2 := (t-3)/4 = 5*t-7; \nt2 := solve(eq2);\nsubs(t=t2,eq2);\nevalb(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 39 "Simultaneous equations in two variables " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 5 "solve" }{TEXT -1 55 " can be \+ used to solve a pair of simultaneous equations." }}{PARA 0 "" 0 "" {TEXT -1 37 "For example, consider the equations: " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([x+3*y = 6 ,`` ],[x-y = 3 , ``])" "6#-%*PIECEWISEG6$7$/,&%\"xG\"\"\"*&\"\"$F*%\"yGF*F*\"\"'%!G7$/ ,&F)F*F-!\"\"F,F/" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "solve(\{x+3*y=6,x-y=3\});" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "The so lution can be illustrated graphically.\nOne way to plot the graphs of \+ the two equations is to use " }{TEXT 0 12 "implicitplot" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "plots[implicitplot](\{x+3*y=6,x-y=3\},x=-1..7,y=-1..5 ,thickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 " 6&-%'CURVESG6O7$7$$!\"\"\"\"!$\"3.LLLLLLLB!#<7$$!3w#***********f$*!#=$ \"3$)***********>J#F-7$7$$!3Q*************z'F1$\"3dmmmmmmEAF-F.7$7$$!3 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-7$7$Fa[l$\"3P,+++++SIF-F^bm7$7$Fa[l$\"3#4++++++/$F-7$$\"31Wr&G9dG1'F- $\"31Wr&G9dG1$F-7$7$$\"3'4++++++3'F-$\"3'4++++++3$F-F[cm7$Facm7$$\"31, ++++++iF-$\"31,++++++KF-7$7$$\"3<,+++++?jF-$\"3<,+++++?LF-Fgcm7$F]dm7$ $\"3'*eG9dG9PjF-$\"3_eG9dG9PLF-7$7$Ff\\l$\"3?,+++++gLF-Fcdm7$Fidm7$$\" 3)fr&G9dGukF-$\"3a:dG9dGuMF-7$7$$\"3Q,+++++glF-$\"3Q,+++++gNF-F]em7$Fc em7$$\"3*HdG9dG9h'F-$\"3as&G9dG9h$F-7$7$Fa]l$\"3[,+++++!o$F-Fiem7$F_fm 7$$\"3!4Vr&G9d[nF-$\"3WI9dG9d[PF-7$7$$\"3g,++++++oF-$\"3g,++++++QF-Fcf m7$Fifm7$$\"3+(G9dG9d)oF-$\"3+(G9dG9d)QF-7$7$Fg]l$\"3y,++++++SF-F_gmFa ^l-%*THICKNESSG6#F[_l-%+AXESLABELSG6$%\"xG%\"yG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "The coordinates of the point of inters ection of the two lines give the solution to the pair of equations." } }{PARA 0 "" 0 "" {TEXT -1 67 "An approximate solution can be obtained \+ from this graph as follows." }}{PARA 15 "" 0 "" {TEXT -1 32 "Select th e graph by clicking it." }}{PARA 15 "" 0 "" {TEXT -1 114 "Place the po int of the arrow cursor as close as you can judge by eye to the locati on of the point of intersection." }}{PARA 15 "" 0 "" {TEXT -1 115 "Rea d off the approximate coordinates for the point of intersection from t he box at the left end of the context bar." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "Another way to plot the graphs \+ is to rearrange the two equations with y as the subject. " }}{PARA 0 " " 0 "" {TEXT -1 68 "In this example it is easy to do this by hand, but you can also use " }{TEXT 0 5 "solve" }{TEXT -1 124 ".\nSince there a re two variables in each equation, we have to specify the variable we \+ want to solve for as a second input or " }{TEXT 259 8 "argument" } {TEXT -1 5 " for " }{TEXT 0 5 "solve" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "solve(x+3* y=6,y);\nsolve(x-y=3,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"xG#! \"\"\"\"$\"\"#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"xG\"\"\" \"\"$!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 125 "Again, it would be easy to set up the functions by hand, but t here is a way to automate this by means of the Maple procedure " } {TEXT 0 7 "unapply" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 14 "Th e procedure " }{TEXT 0 7 "unapply" }{TEXT -1 75 " will convert an expr ession involving one variable, say x, into a function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "f := un apply(-1/3*x+2,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6 \"6$%)operatorG%&arrowGF(,&9$#!\"\"\"\"$\"\"#\"\"\"F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 118 "\nThe process of converting the two equations in to two functions f and g to plot is acheived by the following commands ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "f := unapply(solve(x+3*y=6,y),x);\ng := unapply(solve (x-y=3,y),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$% )operatorG%&arrowGF(,&*&#\"\"\"\"\"$F/9$F/!\"\"\"\"#F/F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&9 $\"\"\"\"\"$!\"\"F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "plot([f(x),g(x)],x=-1..7,y=-1..5,co lor=[red,blue],thickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6$7S7$$!\"\"\"\"!$\"3[LLLLLLLB!#<7$$!3umm mmwAc#)!#=$\"3u)))))))e2_F#F-7$$!3ALLL$o!)*QnF1$\"3PWWW*oKYA#F-7$$!3mm mmmxnK]F1$\"3SbbbDfvn@F-7$$!3#ommmOk]J$F1$\"3\\bbbX@]5@F-7$$!37MLL$[9c g\"F1$\"3+yxx#[?N0#F-7$$!3QBnmm;ct?!#?$\"3RAAA(=\"p+?F-7$$\"3v)*****\\ YJ?;F1$\"3ymmm6&*)f%>F-7$$\"3/KLL$=\"\\F!4'F17$$\"35mmm1>qMVF-$\" 3KYWWWO*4b&F17$$\"3%)*******HSu]%F-$\"3E,+++!*>v\\F17$$\"3'HLL$ep'Rm%F -$\"3#\\AAAZVMX%F17$$\"3')******R>4N[F-$\"3'ymmm'o-$)QF17$$\"3#emm;@2h *\\F-$\"3CZWW%H4jM$F17$$\"3]*****\\c9W;&F-$\"3'\\LLLy%G&y#F17$$\"3Lmmm md'*G`F-$\"3S!yxxx!yOAF17$$\"3j*****\\iN7]&F-$\"3!zmmm\"zai;F17$$\"3aL LLt>:ncF-$\"3-BAAAM\\46F17$$\"35LLL.a#o$eF-$\"3>PAAAK:Ra!#>7$$\"3ammm^ Q40gF-$!3e;_bb0&zp\"FK7$$\"3y******z]rfhF-$!3\"H*******f$QK&Fgx7$$\"3g mmmc%GpL'F-$!3Lbbbb[4B6F17$$\"3/LLL8-V&\\'F-$!3)>WWW/M9l\"F17$$\"3=+++ XhUkmF-$!3#emmm\"Qv9AF17$$\"3=+++:o1DBF-7$F]o$!3kmmmw))yr@F-7 $Fbo$!3;+++S(R#**>F-7$Fgo$!30++++@)f#=F-7$F\\p$!3-+++gi,f;F-7$Fap$!3qm mm\"G&R2:F-7$Ffp$!3XLLLtK5F8F-7$F[q$!3eLLL$HsV<\"F-7$F`q$!3+-++]&)4n** F17$Feq$!37PLLL\\[%R)F17$Fjq$!3G)*****\\&y!pmF17$F_r$!3Y******\\O3E]F1 7$Fdr$!3NKLLL3z6LF17$Fir$!3sLLL$)[`P(** ****z-6j'F17$Fgt$\"3q\"******4#32$)F17$F\\u$\"3r$*****\\#y'G**F17$Fau$ \"3G******H%=H<\"F-7$Ffu$\"35mmm1>qM8F-7$F[v$\"3%)*******HSu]\"F-7$F`v $\"3'HLL$ep'Rm\"F-7$Fev$\"3')******R>4N=F-7$Fjv$\"3#emm;@2h*>F-7$F_w$ \"3]*****\\c9W;#F-7$Fdw$\"3Lmmmmd'*GBF-7$Fiw$\"3j*****\\iN7]#F-7$F^x$ \"3aLLLt>:nEF-7$Fcx$\"35LLL.a#o$GF-7$Fix$\"3ammm^Q40IF-7$F^y$\"3y***** *z]rfJF-7$Fcy$\"3gmmmc%GpL$F-7$Fhy$\"3/LLL8-V&\\$F-7$F]z$\"3=+++XhUkOF -7$Fbz$\"3=+++:o " 0 "" {MPLTEXT 1 0 38 "x1 := solve(f(x)=g(x),x);\ny1 := f(x1);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#x1G#\"#:\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G#\"\"$\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 259 4 "Note" }{TEXT -1 29 ": It is not essential to use \+ " }{TEXT 0 7 "unapply" }{TEXT -1 147 " to convert the expressions to f unctions. It is really just a matter of personal preference whether yo u like working with functions or expressions." }}{PARA 0 "" 0 "" {TEXT -1 113 "I tend to prefer functions most of the time. Expressions are just a bit messy to evaluate, since you need to use " }{TEXT 0 4 "subs" }{TEXT -1 4 " or " }{TEXT 0 4 "eval" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 156 "x := 'x': y := 'y':\nfx := solve(x+3*y=6,y);\ngx := solve(x-y=3,y);\nx1 := solve(fx=gx,x);\ny1 := subs(x=x1,fx);\nplot([fx,gx],x=-1..7,y=-1.. 5,color=[red,blue]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "Quadratic equations" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 30 "We can also use the procedure " } {TEXT 0 5 "solve" }{TEXT -1 30 " to solve quadratic equations." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 9 "Example 1 " }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 15 "Solve for x: " } {XPPEDIT 18 0 "x^2-x-6 = 0;" "6#/,(*$%\"xG\"\"#\"\"\"F&!\"\"\"\"'F)\" \"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "solve(x^2-x-6=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"$!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 264 9 "Example 2" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 15 "Solve for x: " }{XPPEDIT 18 0 "10*x^2+x-21 = 0;" "6#/,( *&\"#5\"\"\"*$%\"xG\"\"#F'F'F)F'\"#@!\"\"\"\"!" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "solve(10*x^2+x-21=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$#\"\"(\" \"&#!\"$\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "If you want decimal answe rs just type . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$ \"+++++9!\"*$!+++++:F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "If you would like to see more details of the soluti on, you can play around with the Maple procedure " }{TEXT 0 6 "factor " }{TEXT -1 30 ". \nFor example, the quadratic " }{XPPEDIT 18 0 "10*x^ 2+x-21;" "6#,(*&\"#5\"\"\"*$%\"xG\"\"#F&F&F(F&\"#@!\"\"" }{TEXT -1 23 " can be factored by . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "factor(10*x^2+x-21);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#*&,&%\"xG\"\"#\"\"$\"\"\"F(,&F%\"\"&\"\"(!\"\"F( " }}}{PARA 0 "" 0 "" {TEXT -1 99 "\nBy equating each of these two fact ors to zero, we can easliy obtain the solutions of the equation " } {XPPEDIT 18 0 "10*x^2+x-21 = 0;" "6#/,(*&\"#5\"\"\"*$%\"xG\"\"#F'F'F)F '\"#@!\"\"\"\"!" }{TEXT -1 9 ", namely " }{XPPEDIT 18 0 "x = 7/5;" "6# /%\"xG*&\"\"(\"\"\"\"\"&!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x = -3/2;" "6#/%\"xG,$*&\"\"$\"\"\"\"\"#!\"\"F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Note that the o pposite to " }{TEXT 0 7 "factor " }{TEXT -1 3 "is " }{TEXT 0 6 "expand " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 24 "expand((5*x-7)*(2*x+3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"xG\"\"#\"\"\"\"#5F&F(\"#@!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "Maple doesn't compla in if you throw in horrible numbers." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 9 "Example 3" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 15 "Solve for x: " }{XPPEDIT 18 0 "7*x^2/34-8*x/3+9/ 11 = 0;" "6#/,(*(\"\"(\"\"\"*$%\"xG\"\"#F'\"#M!\"\"F'*(\"\")F'F)F'\"\" $F,F,*&\"\"*F'\"#6F,F'\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "solve(7/34*x^2-8/3 *x+9/11=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&#\"$O\"\"#@\"\"\"*& #F'\"$J#F'-%%sqrtG6#\"(ef-#F'F',&F$F'*&#F'F*F'*$F+F'F'!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 147 "For quadratics which don't factor \"nicely\", in terms of integers, the usual practi ce when solving quadratic equations by hand is to either use the " } {TEXT 259 17 "quadratic formula" }{TEXT -1 22 " or use the method of \+ " }{TEXT 259 21 "completing the square" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "For example the quad ratic " }{XPPEDIT 18 0 "x^2+6*x+4;" "6#,(*$%\"xG\"\"#\"\"\"*&\"\"'F'F% F'F'\"\"%F'" }{TEXT -1 53 " doen't factor in terms of integers and the equation " }{XPPEDIT 18 0 "x^2+6*x+4 = 0;" "6#/,(*$%\"xG\"\"#\"\"\"*& \"\"'F(F&F(F(\"\"%F(\"\"!" }{TEXT -1 34 " has solutions involving radi cals." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "solve(x^2+6*x+4=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$,&!\"$\"\"\"*$-%%sqrtG6#\"\"&F%F%,&F$F%F&!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "We can get Maple can " } {TEXT 259 14 "show the steps" }{TEXT -1 72 " of solving a quadratic eq uation by the method of completing the square." }}{PARA 0 "" 0 "" {TEXT -1 15 "First load the " }{TEXT 0 7 "student" }{TEXT -1 44 " pack age, which contains a procedure called " }{TEXT 0 14 "completesquare" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(student);" }}{PARA 12 "" 1 "" {XPPMATH 20 " 6#7@%\"DG%%DiffG%*DoubleintG%$IntG%&LimitG%(LineintG%(ProductG%$SumG%* TripleintG%*changevarG%/completesquareG%)distanceG%'equateG%*integrand G%*interceptG%)intpartsG%(leftboxG%(leftsumG%)makeprocG%*middleboxG%*m iddlesumG%)midpointG%(powsubsG%)rightboxG%)rightsumG%,showtangentG%(si mpsonG%&slopeG%(summandG%*trapezoidG" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 15 "Information on " }{TEXT 0 14 "complet esquare" }{TEXT -1 25 " can be obtained from . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "?completesqu are" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "T his procedure can be applied to a quadratic expression . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "comp letesquare(x^2+6*x+4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$),&%\"xG \"\"\"\"\"$F(\"\"#F(F(\"\"&!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 25 ". . or to an equation . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "complet esquare(x^2+6*x+4=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*$),&%\"xG \"\"\"\"\"$F)\"\"#F)F)\"\"&!\"\"\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 74 "To procede towards a solution from he re requires isolating the expression " }{XPPEDIT 18 0 "(x+3)^2;" "6#*$ ,&%\"xG\"\"\"\"\"$F&\"\"#" }{TEXT -1 123 " on the left side of the equ ation and putting the constant -5 on the right side as 5. (Add 5 to ea ch side to achieve this.)" }}{PARA 0 "" 0 "" {TEXT -1 66 "The followin g commands do some pattern matching using the routine " }{TEXT 0 8 "pa tmatch" }{TEXT -1 28 " to extract the bits needed." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 180 "comps := co mpletesquare(x^2+6*x+4);\nexpr := 'expr': const := 'const':\npatmatch( comps,(expr::algebraic)^2+const::realcons,'params'):\nexpr := rhs(para ms[1]);\nconst := rhs(params[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %&compsG,&*$),&%\"xG\"\"\"\"\"$F*\"\"#F*F*\"\"&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%exprG,&%\"xG\"\"\"\"\"$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&constG!\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 76 "The solution is given by taking square ro ots on both sides of the equation " }{XPPEDIT 18 0 "(x+3)^2 = 5;" "6# /*$,&%\"xG\"\"\"\"\"$F'\"\"#\"\"&" }{TEXT -1 10 " which is " } {XPPEDIT 18 0 "expr^2 = const;" "6#/*$%%exprG\"\"#%&constG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "expr^2 = -const;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/*$),&%\"xG\"\"\"\"\"$F(\"\"#F(\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 26 "Taking square roots gives:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "ex pr = sqrt(-const);\nexpr = -sqrt(-const);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"xG\"\"\"\"\"$F&*$-%%sqrtG6#\"\"&F&" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/,&%\"xG\"\"\"\"\"$F&,$*$-%%sqrtG6#\"\"&F&!\"\" " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "Let' s be lazy by bringing out a sledgehammer to crack a nutshell and use \+ " }{TEXT 0 5 "solve" }{TEXT -1 33 " to write down the two solutions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "solve(expr=sqrt(-const),x);\nsolve(expr=-sqrt(-const),x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&!\"$\"\"\"*$-%%sqrtG6#\"\"&F%F%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&!\"$\"\"\"*$-%%sqrtG6#\"\"&F%!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "We now put this all together." }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 1 ":" }}{PARA 15 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 14 " completesquare" }{TEXT -1 69 " can be accessed without loading the who le student package by typing " }{TEXT 0 23 "student[completesquare]" } {TEXT -1 137 ".\nIf you have followed through this worksheet step by s tep, the student package will already be loaded, and it is only necess ary to type " }{TEXT 0 14 "completesquare" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 109 "The output is controlled by suppressing the resu lts of the calculations and printing corresponding equations." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 398 "x := 'x':\neq := x^2+6*x+4 =0: print(eq);\ncomps := student[compl etesquare](lhs(eq)): print(comps=0);\nexpr := 'expr': const := 'const' :\npatmatch(comps,(expr::algebraic)^2+const::realcons,'params'):\nexpr := rhs(params[1]):\nconst := rhs(params[2]):\nprint(expr^2=-const);\n print(expr=sqrt(-const),expr=-sqrt(-const));\nx1 := solve(expr=sqrt(-c onst),x):\nx2 := solve(expr=-sqrt(-const),x):\nprint(x=x1,x=x2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*$)%\"xG\"\"#\"\"\"F)*&\"\"'F)F'F)F )\"\"%F)\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*$),&%\"xG\"\"\"\" \"$F)\"\"#F)F)\"\"&!\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$), &%\"xG\"\"\"\"\"$F(\"\"#F(\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/,& %\"xG\"\"\"\"\"$F&*$-%%sqrtG6#\"\"&F&/F$,$F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%\"xG,&!\"$\"\"\"*$-%%sqrtG6#\"\"&F'F'/F$,&F&F'F(!\"\" " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 109 "The same block of instructions can be used to illustrate the solution of \+ any quadratic equation of the form " }{XPPEDIT 18 0 "x^2+b*x-c=0" "6# /,(*$%\"xG\"\"#\"\"\"*&%\"bGF(F&F(F(%\"cG!\"\"\"\"!" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 388 "x := 'x':\neq := x^2+7*x+2=0:\nprint(eq);\ncomps := completesqu are(lhs(eq)): print(comps=0);\nexpr := 'expr': const := 'const':\npatm atch(comps,(expr::algebraic)^2+const::realcons,'params'):\nexpr := rhs (params[1]):\nconst := rhs(params[2]):\nprint(expr^2=-const);\nprint(e xpr=sqrt(-const),expr=-sqrt(-const));\nx1 := solve(expr=sqrt(-const),x ):\nx2 := solve(expr=-sqrt(-const),x):\nprint(x=x1,x=x2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*$)%\"xG\"\"#\"\"\"F)*&\"\"(F)F'F)F)F(F)\"\" !" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*$),&%\"xG\"\"\"#\"\"(\"\"#F)F ,F)F)#\"#T\"\"%!\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$),&%\" xG\"\"\"#\"\"(\"\"#F(F+F(#\"#T\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $/,&%\"xG\"\"\"#\"\"(\"\"#F&,$*$-%%sqrtG6#\"#TF&#F&F)/F$,$F+#!\"\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%\"xG,&#!\"(\"\"#\"\"\"*&#F)F(F)-%% sqrtG6#\"#TF)F)/F$,&F&F)*&#F)F(F)*$F,F)F)!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 5 "solve" }{TEXT -1 23 " gives the same answer." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "solve(eq);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $,&#!\"(\"\"#\"\"\"*&#F'F&F'-%%sqrtG6#\"#TF'F',&F$F'*&#F'F&F'*$F*F'F'! \"\"" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 178 "With a couple of modifications (spot them), our instructions can be used to illustrate the derivation of a simplfied quadratic formula for the solution of the quadratic equatio n " }{XPPEDIT 18 0 "x^2+b*x+c = 0;" "6#/,(*$%\"xG\"\"#\"\"\"*&%\"bGF(F &F(F(%\"cGF(\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 391 "x := 'x':\neq := x^2+b*x+c= 0:\nprint(eq);\ncomps := student[completesquare](lhs(eq),x): print(com ps=0);\nexpr := 'expr': const := 'const':\npatmatch(comps,(expr::algeb raic)^2+const::algebraic,'params'):\nexpr := rhs(params[1]):\nconst := rhs(params[2]):\nprint(expr^2=-const);\nsq := simplify(sqrt(-const)): \nprint(expr=sq,expr=-sq);\nx1 := solve(expr=sq,x):\nx2 := solve(expr =-sq,x):\nprint(x=x1,x=x2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*$)% \"xG\"\"#\"\"\"F)*&%\"bGF)F'F)F)%\"cGF)\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*$),&%\"xG\"\"\"*&#F)\"\"#F)%\"bGF)F)F,F)F)*&#F)\"\" %F)*$)F-F,F)F)!\"\"%\"cGF)\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$ ),&%\"xG\"\"\"*&#F(\"\"#F(%\"bGF(F(F+F(,&*$)F,F+F(#F(\"\"%%\"cG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/,&%\"xG\"\"\"*&#F&\"\"#F&%\"bGF&F&, $*$-%%sqrtG6#,&*$)F*F)F&F&*&\"\"%F&%\"cGF&!\"\"F&F(/F$,$F,#F6F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$/%\"xG,&%\"bG#!\"\"\"\"#*&#\"\"\"F)F,- %%sqrtG6#,&*$)F&F)F,F,*&\"\"%F,%\"cGF,F(F,F,/F$,&F&F'*&#F,F)F,*$F-F,F, F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 5 "solve" }{TEXT -1 54 " will \+ give the quadratic formula for the solutions of " }{XPPEDIT 18 0 "a*x^ 2+b*x+c = 0;" "6#/,(*&%\"aG\"\"\"*$%\"xG\"\"#F'F'*&%\"bGF'F)F'F'%\"cGF '\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "solve(a*x^2+b*x+c = 0,x);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 115 "Here is \+ way of demonstrating a derivation of the quadratic formula with Maple \+ doing most, but not all, of the work." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 117 ": You must execu te the following sequence of commands in the order in which they appea r, because the single variable " }{TEXT 0 2 "eq" }{TEXT -1 52 " is use d to store the current equation at each step." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "Set up the equation " } {XPPEDIT 18 0 "a*x^2+b*x+c = 0;" "6#/,(*&%\"aG\"\"\"*$%\"xG\"\"#F'F'*& %\"bGF'F)F'F'%\"cGF'\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "a := 'a': b := 'b': c := 'c': x := 'x':\neq := a*x^2 + b*x + c = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eqG/,(*&%\"aG\"\"\")%\"xG\"\"#F)F)*&%\"bGF)F+F)F)%\" cGF)\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Multiply the equation by 4a." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "eq := expand(4*a*eq);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eqG/,(*&)%\"aG\"\"#\"\"\")%\"xGF*F+ \"\"%**F.F+F)F+%\"bGF+F-F+F+*(F.F+F)F+%\"cGF+F+\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Subtract " }{XPPEDIT 18 0 "4*a*c;" "6#*(\"\"%\"\"\"%\"aGF%%\"cGF%" }{TEXT -1 43 " from both sides of the equation (by hand)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "eq := lhs(eq) - 4*a*c = -4*a *c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eqG/,&*&)%\"aG\"\"#\"\"\")% \"xGF*F+\"\"%**F.F+F)F+%\"bGF+F-F+F+,$*&F)F+%\"cGF+!\"%" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Add " }{XPPEDIT 18 0 "b^2;" "6#*$%\"bG\"\"#" }{TEXT -1 15 " to both sides." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "eq :=lh s(eq)+b^2 = rhs(eq)+b^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eqG/,(* &)%\"aG\"\"#\"\"\")%\"xGF*F+\"\"%**F.F+F)F+%\"bGF+F-F+F+*$)F0F*F+F+,&* &F)F+%\"cGF+!\"%F1F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Factor the left side, which is now a perfect square." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "eq := factor(lhs(eq))=rhs(eq);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#eqG/*$),&*&%\"aG\"\"\"%\"xGF+\"\"#%\"bGF+F-F+,&*&F*F+%\"cGF+!\"%* $)F.F-F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Bring out " }{TEXT 0 5 "solve" }{TEXT -1 101 " to do the last s tep of taking square roots and solving each of the resulting linear eq uations for x." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "solve(eq,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ ,$*&,&%\"bG!\"\"*$-%%sqrtG6#,&*&%\"aG\"\"\"%\"cGF/!\"%*$)F&\"\"#F/F/F/ F/F/F.F'#F/F4,$*&,&F&F'F(F'F/F.F'F5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 "Cubic equations" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 87 "Cubics without a qu adratic term could be solved by Cardano and Tartaglia in the 1400's." }}{PARA 0 "" 0 "" {TEXT -1 85 "A cubic equation with real coefficients always has at least one real number solution." }}{PARA 0 "" 0 "" {TEXT -1 28 "In the case of the equation " }{XPPEDIT 18 0 "x^3+p*x+q = 0;" "6#/,(*$%\"xG\"\"$\"\"\"*&%\"pGF(F&F(F(%\"qGF(\"\"!" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "27*q^2>4*p^3" "6#2*&\"\"%\"\"\"*$%\"pG\"\"$ F&*&\"#FF&*$%\"qG\"\"#F&" }{TEXT -1 26 ", we have the solution . ." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "unassign('x','p','q');\neq := x^3 + p*x + q = 0;\n[solve(eq,x)]:\n x0 := op(remove(has,%,Complex(1)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#eqG/,(*$)%\"xG\"\"$\"\"\"F+*&%\"pGF+F)F+F+%\"qGF+\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G,&*$),&%\"qG!$3\"*&\"#7\"\"\"-%%sqrtG6 #,&*$)%\"pG\"\"$F-F,*&\"#\")F-)F)\"\"#F-F-F-F-#F-F5F-#F-\"\"'*(F9F-F4F -F(#!\"\"F5F?" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 16 ": The procedure " }{TEXT 0 6 "remove " }{TEXT -1 56 " removes any solutions which involve the imaginary uni t " }{TEXT 0 1 "I" }{TEXT -1 17 " (referred to by " }{TEXT 0 10 "Compl ex(1)" }{TEXT -1 72 "), that is, it removes any solutions which are no n-real complex numbers." }}{PARA 0 "" 0 "" {TEXT -1 41 "To check the s olution, we can substitute " }{XPPEDIT 18 0 "x = x[0];" "6#/%\"xG&F$6# \"\"!" }{TEXT -1 7 " into " }{XPPEDIT 18 0 "x^3+p*x+q;" "6#,(*$%\"xG \"\"$\"\"\"*&%\"pGF'F%F'F'%\"qGF'" }{TEXT -1 4 " . ." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "subs(x=x0, x^3+p*x+q);\nsimplify(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,(*$),&*$ ),&%\"qG!$3\"*&\"#7\"\"\"-%%sqrtG6#,&*$)%\"pG\"\"$F.F-*&\"#\")F.)F*\" \"#F.F.F.F.#F.F6F.#F.\"\"'*&*&F:F.F5F.F.*$)F)#F.F6F.!\"\"FCF6F.F.*&F5F .F&F.F.F*F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{PARA 0 "" 0 "" {TEXT -1 28 "Consider the cubic equation " }{XPPEDIT 18 0 "x^3-x-1 = 0 ;" "6#/,(*$%\"xG\"\"$\"\"\"F&!\"\"F(F)\"\"!" }{TEXT -1 4 " or " } {XPPEDIT 18 0 "x^3 = x+1;" "6#/*$%\"xG\"\"$,&F%\"\"\"F(F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "eq1 := x^3-x-1=0;\n[solve(eq1,x)]:\nop(remove(has,%,C omplex(1)));\nx1 := evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq 1G/,(*$)%\"xG\"\"$\"\"\"F+F)!\"\"F+F,\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$),&\"$3\"\"\"\"*&\"#7F(-%%sqrtG6#\"#pF(F(#F(\"\"$F( #F(\"\"'*&\"\"#F(F&#!\"\"F0F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1 G$\"+ezrC8!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "We can check that " }{XPPEDIT 18 0 "x[1]^3;" "6#*$&%\"xG6 #\"\"\"\"\"$" }{TEXT -1 27 " is approximately equal to " }{XPPEDIT 18 0 "x[1]+1;" "6#,&&%\"xG6#\"\"\"F'F'F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "x1 + 1;\n x1^3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ezrCB!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+hzrCB!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 77 "The solution is illustrated graphically a s the intersection of the graphs of " }{XPPEDIT 18 0 "y = x^3;" "6#/% \"yG*$%\"xG\"\"$" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = x+1;" "6#/% \"yG,&%\"xG\"\"\"F'F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "T he x" }{TEXT 266 0 "" }{TEXT -1 65 " coordinate of the point of inters ection is the real solution of " }{XPPEDIT 18 0 "x^3 = x+1;" "6#/*$%\" xG\"\"$,&F%\"\"\"F(F(" }{TEXT -1 10 ", and the " }{TEXT 267 1 "y" } {TEXT -1 36 " coordinate is the common value of " }{XPPEDIT 18 0 "x^3 ;" "6#*$%\"xG\"\"$" }{TEXT -1 5 " and " 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Qj\"F*7$F]v$\"3?+++?h(Hq\"F*7$Fbv$\"3?LLL$y'elF*7$Ffw$\"3immm1jeJ?F*7$F[x$\" 3n******\\U\\+@F*7$F`x$\"3eLLL*ygo;#F*7$Fex$\"3DLLLh,tMAF*7$Fjx$\"3)om m1aP?I#F*7$F_y$\"3#******>.')QO#F*7$Fdy$\"3#ommEQrZV#F*7$Fiy$\"37LLL&3 s\")\\#F*7$F^z$\"3E+++e/xlDF*7$Fcz$\"3)******fsq/j#F*7$Fhz$\"3;+++++++ FF*-F][l6&F_[lFc[lFc[lF`[l-%*THICKNESSG6#\"\"#-%+AXESLABELSG6$Q\"x6\"Q \"yFeel-%%VIEWG6$;$!#:!\"\"$\"# " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 " }}{PARA 0 "" 0 "" {TEXT -1 28 "Consider the cubic equation " }{XPPEDIT 18 0 "x^3+2*x-2 = 0;" "6#/,(*$%\"xG\"\"$\"\"\"*&\"\"#F(F&F(F(F*!\"\"\"\"!" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "x^3 = -2*x+2;" "6#/*$%\"xG\"\"$,&*&\"\"#\"\"\"F %F*!\"\"F)F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "eq2 := x^3+2*x-2=0;\n[solve( eq2,x)]:\nop(remove(has,%,Complex(1)));\nx2 := evalf(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%$eq2G/,(*$)%\"xG\"\"$\"\"\"F+*&\"\"#F+F)F+F+F- !\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$),&\"#F\"\"\"*&\"\"$ F(-%%sqrtG6#\"$0\"F(F(#F(F*F(F/*&\"\"#F(F&#!\"\"F*F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x2G$\"+u*p\"4x!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "We can check that " }{XPPEDIT 18 0 " x[2]^3;" "6#*$&%\"xG6#\"\"#\"\"$" }{TEXT -1 27 " is approximately equa l to " }{XPPEDIT 18 0 "-2*x[2]+2;" "6#,&*&\"\"#\"\"\"&%\"xG6#F%F&!\"\" F%F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "-2*x2 + 2;\nx2^3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"*0g;e%!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+l+ m\"e%!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "The solution is illustrated graphically as the intersection of the graphs of " }{XPPEDIT 18 0 "y = x^3;" "6#/%\"yG*$%\"xG\"\"$" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = -2*x+2;" "6#/%\"yG,&*&\"\"#\"\"\"%\"x GF(!\"\"F'F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 268 1 "x" }{TEXT -1 65 " coordinate 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" }}{PARA 0 "" 0 "" {TEXT -1 5 "(a) " }{XPPEDIT 18 0 "x/2+6 = 2*x;" "6#/,&*&%\"xG\"\"\"\"\"#!\" \"F'\"\"'F'*&F(F'F&F'" }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "(b) " }{XPPEDIT 18 0 "x/5+2 = (15+x)/10;" "6#/,&*&%\"xG\"\"\"\"\"&! \"\"F'\"\"#F'*&,&\"#:F'F&F'F'\"#5F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 6 "(c) " }{XPPEDIT 18 0 "3*x/7-5/21 = (2-x)/14;" "6#/,&*( \"\"$\"\"\"%\"xGF'\"\"(!\"\"F'*&\"\"&F'\"#@F*F**&,&\"\"#F'F(F*F'\"#9F* " }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 7 "(d) " }{XPPEDIT 18 0 "(x-3)/12-2/3 = (1-3*x)/2;" "6#/,&*&,&%\"xG\"\"\"\"\"$!\"\"F(\"#7 F*F(*&\"\"#F(F)F*F**&,&F(F(*&F)F(F'F(F*F(F-F*" }{TEXT -1 3 " " }} {PARA 0 "" 0 "" {TEXT -1 34 "_________________________________ " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 34 "_________________________________ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 109 " In quest ions 2 to 7, solve the given pair of simultaneous equations and illust rate the solution graphically." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 " Q2 " }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "PIECEWISE([y = -x+4, ``],[y = x-2, ``]);" "6#-%*PIECEWISEG6$7$/%\"yG,&%\"xG!\"\"\"\" %\"\"\"%!G7$/F(,&F*F-\"\"#F+F." }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 34 "_________________________________ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "__ _______________________________ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q3 " }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "PIECEWISE([3*x+2*y = 6, ``],[x-3*y = 3, ``]);" "6#-%*PIECEWISEG6$7$/,&*&\"\"$\"\"\"%\"xGF+F+*&\"\"#F+%\" yGF+F+\"\"'%!G7$/,&F,F+*&F*F+F/F+!\"\"F*F1" }{TEXT -1 4 " " }} {PARA 0 "" 0 "" {TEXT -1 34 "_________________________________ " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 34 "_________________________________ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q4 \+ " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "PIECEWISE([2*y-2*x = 7, ``],[4*x-2*y = 1, ``]);" "6#-%*PIECEWISEG6$7$/,&*&\"\"#\"\"\"%\" yGF+F+*&F*F+%\"xGF+!\"\"\"\"(%!G7$/,&*&\"\"%F+F.F+F+*&F*F+F,F+F/F+F1" }{TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT -1 34 "___________________ ______________ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 34 "_________________________________ " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "Q5 " }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "PIECEWISE([5*x-y = 3, ``],[2*y-4*x = 3, ``]);" "6#-%*PIECEWISEG6$7$/, &*&\"\"&\"\"\"%\"xGF+F+%\"yG!\"\"\"\"$%!G7$/,&*&\"\"#F+F-F+F+*&\"\"%F+ F,F+F.F/F0" }{TEXT -1 6 " " }}{PARA 0 "" 0 "" {TEXT -1 34 "______ ___________________________ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "____________________ _____________ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 57 "In questions 6 to 9, solve the given quad ratic equation. " }}{PARA 0 "" 0 "" {TEXT -1 121 "In each case determi ne whether the left-hand side factors in terms integers, and in any su ch case give the factorization." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q6 " }}{PARA 0 "" 0 "" {TEXT -1 6 "(a) " }{XPPEDIT 18 0 "x^2-10*x+1 6 = 0;" "6#/,(*$%\"xG\"\"#\"\"\"*&\"#5F(F&F(!\"\"\"#;F(\"\"!" }{TEXT -1 9 " " }}{PARA 0 "" 0 "" {TEXT -1 34 "______________________ ___________ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 34 "_________________________________ " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "Q7 " }}{PARA 0 "" 0 "" {TEXT -1 5 "(b) " }{XPPEDIT 18 0 "3*x^2+11*x-4;" "6#,(*&\"\"$\"\"\"*$%\"xG\"\"#F&F&*&\"#6F&F(F&F&\"\" %!\"\"" }{TEXT -1 12 " " }}{PARA 0 "" 0 "" {TEXT -1 34 "___ ______________________________ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "____________________ _____________ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q8 " }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "4*x^2-3*x-2 = 0;" "6#/,(*&\"\"%\"\"\"*$%\"xG\"\" #F'F'*&\"\"$F'F)F'!\"\"F*F-\"\"!" }{TEXT -1 7 " (c) " }{XPPEDIT 18 0 "x^2+2*x-5 = 0;" "6#/,(*$%\"xG\"\"#\"\"\"*&F'F(F&F(F(\"\"&!\"\"\"\"! " }{TEXT -1 9 " " }}{PARA 0 "" 0 "" {TEXT -1 34 "_____________ ____________________ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "_______________________________ __ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q9 " }}{PARA 0 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "4*x^2-3*x-2 = 0;" "6#/,(*&\"\"%\"\"\"*$%\"xG\"\"#F'F'*& \"\"$F'F)F'!\"\"F*F-\"\"!" }{TEXT -1 9 " " }}{PARA 0 "" 0 "" {TEXT -1 34 "_________________________________ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "__ _______________________________ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 104 "In questions 10 and 11, fi nd the all real number solutions of each of the given cubic equation b y using " }{TEXT 0 5 "solve" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 58 "Check your solutions graphically by plotting the graph of " } {XPPEDIT 18 0 "y = x^3;" "6#/%\"yG*$%\"xG\"\"$" }{TEXT -1 40 " togethe r with a suitable straight line." }}{PARA 0 "" 0 "" {TEXT -1 35 "Try f actoring the cubic expression." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 " Q10 " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x^3-x-2 = 0;" "6#/,(*$%\"xG\"\"$\"\"\"F&!\"\"\"\"#F)\"\"!" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 34 "_________________________________ " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "_________________________________ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "Q11 " }} {PARA 0 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "x^3-2*x-1;" "6#,(*$% \"xG\"\"$\"\"\"*&\"\"#F'F%F'!\"\"F'F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 34 "_________________________________ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "__ _______________________________ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "\004" }}}}{MARK "4 0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }