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{SECT 0 {PARA 0 "" 0 "" {TEXT 284 49 "High School Modules > Algebra by
 Gregory A. Moore" }}{PARA 3 "" 0 "" {TEXT -1 4 "    " }{TEXT 283 28 "
Solving Squareroot Equations" }}{PARA 0 "" 0 "" {TEXT -1 94 "\nSolving
 equations with squareroots requires additional steps and the need to \+
check answers. \n" }}{PARA 0 "" 0 "" {TEXT 285 154 "[Directions : Exec
ute the Code Resource section first. Although there will be no output \+
immediately, these definitions are used later in this worksheet.]\n" }
}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 7 "0. Code" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 22 "restart; with(plots): " }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 81 "AddBothSides  :=  proc( expr, a )\n    lhs(expr) + \+
a = rhs( expr) + a; \n end proc:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 104 "MultBothSides  :=  proc( expr, a )\n    simplify(lhs
(expr) * a) = simplify( rhs( expr) * a) ; \n end proc:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "SimplifyBothSides := proc( expr )
\n    simplify( expand( lhs(expr) )) = simplify( expand( rhs(expr) ));
\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "FactorBothSi
des := proc( expr )\n    factor( lhs(expr) ) = factor( rhs(expr) );\ne
nd proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "SquareBothSides
 := proc( expr )\n    simplify( lhs(expr)^2 ) = simplify( rhs(expr)^2 \+
);\nend proc:" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 52 "1. Example 1 :
 Solving Equations With One Squareroot" }}{PARA 0 "" 0 "" {TEXT -1 38 
"\nHere is a straight forward problem.\n\n" }{TEXT 256 7 "Problem" }
{TEXT -1 19 " :  Solve for x :  " }{XPPEDIT 18 0 "sqrt( x + 3) = x - 2
" "6#/-%%sqrtG6#,&%\"xG\"\"\"\"\"$F),&F(F)\"\"#!\"\"" }{TEXT -1 2 ".\n
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Eq := \n       sqrt( x +
 3) = x - 2;" }}}{PARA 0 "" 0 "" {TEXT -1 11 "           " }{TEXT 257 
7 " Step 1" }{TEXT -1 63 ". Since the square root is already isolated,
 square both sides." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "Squar
eBothSides( %);" }}}{PARA 0 "" 0 "" {TEXT -1 12 " \n          " }
{TEXT 258 6 "Step 2" }{TEXT -1 21 ". Expand and simplify" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "SimplifyBothSides( %);" }}}{PARA 0 
"" 0 "" {TEXT -1 11 "\n          " }{TEXT 259 6 "Step 3" }{TEXT -1 29 
". Move everything to one side" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 27 "AddBothSides( %, -lhs(%) );" }}}{PARA 0 "" 0 "" {TEXT -1 12 "\n \+
          " }{TEXT 260 6 "Step 4" }{TEXT -1 32 ". Solve the quadratic \+
equation. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "soln :=  solve
(%, x);" }}}{PARA 0 "" 0 "" {TEXT -1 12 "\n           " }{TEXT 261 6 "
Step 5" }{TEXT -1 47 ". The Often Forgotten Step - Check the answers!
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "subs( x = soln[1], Eq); \+
 \nSimplifyBothSides( %); \ntesteq( lhs(%) = rhs(%));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "subs( x = soln[2], Eq);  \nSimplify
BothSides( %); \ntesteq( lhs(%) = rhs(%));" }}}{PARA 0 "" 0 "" {TEXT 
-1 98 "\nThe first solution fails, while the second works out. Therefo
re the solution to this problem is :" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "Answer := soln[2];" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 
52 "2. Example 2 : Solving Equations With One Squareroot" }}{PARA 0 "
" 0 "" {TEXT -1 38 "\nHere is a straight forward problem.\n\n" }{TEXT 
262 7 "Problem" }{TEXT -1 17 " :  Solve for x :" }{MPLTEXT 1 0 1 " " }
{XPPEDIT 18 0 "3*x - sqrt( 2*x + 1) = 4*x - 9" "6#/,&*&\"\"$\"\"\"%\"x
GF'F'-%%sqrtG6#,&*&\"\"#F'F(F'F'F'F'!\"\",&*&\"\"%F'F(F'F'\"\"*F/" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "Eq := \n       3*x - sqrt( 2
*x + 1) = 4*x - 9;" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "
" {TEXT -1 12 " \n          " }{TEXT 273 6 "Step 1" }{TEXT -1 25 ". Is
olate the square root" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Add
BothSides( %, -3*x);" }}}{PARA 0 "" 0 "" {TEXT -1 10 "          " }
{TEXT 263 7 " Step 2" }{TEXT -1 20 ". Square both sides." }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "SquareBothSides( %);" }}}{PARA 0 "
" 0 "" {TEXT -1 12 " \n          " }{TEXT 264 6 "Step 3" }{TEXT -1 21 
". Expand and simplify" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Si
mplifyBothSides( %);" }}}{PARA 0 "" 0 "" {TEXT -1 11 "\n          " }
{TEXT 265 6 "Step 4" }{TEXT -1 29 ". Move everything to one side" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "AddBothSides( %, -lhs(%) );
" }}}{PARA 0 "" 0 "" {TEXT -1 12 "\n           " }{TEXT 266 6 "Step 5
" }{TEXT -1 32 ". Solve the quadratic Equation. " }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 21 "soln :=  solve(%, x);" }}}{PARA 0 "" 0 "" 
{TEXT -1 12 "\n           " }{TEXT 267 6 "Step 6" }{TEXT -1 61 ". The \+
Often Forgotten Step - Check the answers! It's possible" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "subs( x = soln[1], Eq);  \nSimplify
BothSides( %); \ntesteq( lhs(%) = rhs(%));" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 75 "subs( x = soln[2], Eq);  \nSimplifyBothSides( %); \+
\ntesteq( lhs(%) = rhs(%));" }}}{PARA 0 "" 0 "" {TEXT -1 111 "\nIn thi
s case, the first solution succeeds, while the second fails. Therefore
 the solution to this problem is :" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "Answer := soln[1];" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{SECT 0 {PARA 4 "" 0 "" {TEXT -1 53 "3. Example 3 : Solving Equations \+
With Two Squareroots" }}{PARA 0 "" 0 "" {TEXT -1 165 "\nThe plot thick
ens when there are two square roots. Basically, we need to square twic
e, \nbut being careful to isolate the square roots on one side the sec
ond time.\n\n" }{TEXT 268 7 "Problem" }{TEXT -1 19 " :  Solve for x : \+
 " }{XPPEDIT 18 0 "sqrt(2*x+3)-sqrt(2*x-3) = 4;" "6#/,&-%%sqrtG6#,&*&
\"\"#\"\"\"%\"xGF+F+\"\"$F+F+-F&6#,&*&F*F+F,F+F+F-!\"\"F2\"\"%" }
{TEXT -1 2 ".\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Eq := \n \+
      sqrt( 2*x + 3) - sqrt( 2*x - 3) = 4;" }}}{PARA 0 "" 0 "" {TEXT 
-1 11 "           " }{TEXT 269 7 " Step 1" }{TEXT -1 63 ". Since the s
quare root is already isolated, square both sides." }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 43 "SquareBothSides( %);\nSimplifyBothSides( %);
" }}}{PARA 0 "" 0 "" {TEXT -1 11 "\n          " }{TEXT 270 7 " Step 2
" }{TEXT -1 29 ". Move everything to one side" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 24 "AddBothSides( %, -4*x );" }}}{PARA 0 "" 0 "" 
{TEXT -1 11 " \n         " }{TEXT 274 8 "  Step 3" }{TEXT -1 51 ". Now
, square both sides a second time, and expand." }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 43 "SquareBothSides( %);\nSimplifyBothSides( %);" }}
}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 7 "       \+
" }{TEXT 275 8 "  Step 4" }{TEXT -1 39 ". Move everything to one side \+
to solve." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "AddBothSides( %
, -lhs(%) );" }}}{PARA 0 "" 0 "" {TEXT -1 9 "\n        " }{TEXT 271 6 
"Step 5" }{TEXT -1 32 ". Solve the quadratic Equation. " }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "soln :=  solve(%, x);" }}}{PARA 0 "
" 0 "" {TEXT -1 10 "\n         " }{TEXT 272 6 "Step 6" }{TEXT -1 46 ".
 The Often Forgotten Step - Check the answer." }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 72 "subs( x = soln, Eq);  \nSimplifyBothSides( %); \+
\ntesteq( lhs(%) = rhs(%));" }}}{PARA 0 "" 0 "" {TEXT -1 82 "\nThe onl
y solution fails, so this means that there is no solution to this prob
lem!" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Answer := \{\};" }}}
{PARA 0 "" 0 "" {TEXT -1 202 "\nHow can this be? Lets examine the orig
inal equation in more detail. If we graph the left and right hand side
s of the original equation, we see the two never intersect. This is wh
y there is no solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "p
lot( \{ lhs(Eq), rhs(Eq)\}, x = 1.5..100);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "
" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 
0 "" {TEXT -1 53 "4. Example 4 : Solving Equations With Two Squareroot
s" }}{PARA 0 "" 0 "" {TEXT -1 165 "\nThe plot thickens when there are \+
two square roots. Basically, we need to square twice, \nbut being care
ful to isolate the square roots on one side the second time.\n\n" }
{TEXT 276 7 "Problem" }{TEXT -1 19 " :  Solve for x :  " }{XPPEDIT 18 
0 "sqrt(2*x+3)-sqrt(2*x-3) = 4;" "6#/,&-%%sqrtG6#,&*&\"\"#\"\"\"%\"xGF
+F+\"\"$F+F+-F&6#,&*&F*F+F,F+F+F-!\"\"F2\"\"%" }{TEXT -1 2 ".\n" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Eq := \n       sqrt( 7 - x) \+
+  sqrt( 7 + x) = x;" }}}{PARA 0 "" 0 "" {TEXT -1 11 "           " }
{TEXT 277 7 " Step 1" }{TEXT -1 63 ". Since the square root is already
 isolated, square both sides." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 43 "SquareBothSides( %);\nSimplifyBothSides( %);" }}}{PARA 0 "" 0 "
" {TEXT -1 11 "\n          " }{TEXT 278 7 " Step 2" }{TEXT -1 29 ". Mo
ve everything to one side" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 
"AddBothSides( %, -14 );" }}}{PARA 0 "" 0 "" {TEXT -1 11 " \n         \+
" }{TEXT 281 8 "  Step 3" }{TEXT -1 51 ". Now, square both sides a sec
ond time, and expand." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Squ
areBothSides( %);\nSimplifyBothSides( %);" }}}{PARA 0 "" 0 "" {TEXT 
-1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 7 "       " }{TEXT 282 8 "  Step 4
" }{TEXT -1 39 ". Move everything to one side to solve." }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "AddBothSides( %, -lhs(%) );\nFactor
BothSides( %);" }}}{PARA 0 "" 0 "" {TEXT -1 9 "\n        " }{TEXT 279 
6 "Step 5" }{TEXT -1 52 ". Solve the equation. There are three unique \+
ansers." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "soln :=  solve(%,
 x);" }}}{PARA 0 "" 0 "" {TEXT -1 10 "\n         " }{TEXT 280 6 "Step \+
6" }{TEXT -1 20 ". Check the answers." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 75 "subs( x = soln[1], Eq);  \nSimplifyBothSides( %); \nt
esteq( lhs(%) = rhs(%));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 
"subs( x = soln[2], Eq);  \nSimplifyBothSides( %); \ntesteq( lhs(%) = \+
rhs(%));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "subs( x = soln[
3], Eq);  \nSimplifyBothSides( %); \ntesteq( lhs(%) = rhs(%));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "subs( x = soln[4], Eq);  \nS
implifyBothSides( %); \ntesteq( lhs(%) = rhs(%));" }}}{PARA 0 "" 0 "" 
{TEXT -1 28 "\nThere is only one solution." }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 18 "Answer := soln[3];" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" 
{TEXT 286 36 "\n         \251 2002 Waterloo Maple Inc " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{MARK "0 0" 29 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 
1 }{PAGENUMBERS 0 1 2 33 1 1 }