View of /trunk/NationalProblemLibrary/Utah/Intermediate_Algebra/set8_Quadratic_Equations_Functions_and_Inequalities/s8p12.pg

    1 ##DESCRIPTION
    2 ##Arithmetic
    3 ##ENDDESCRIPTION
    4 ## Author('Utah ww group')
    5 ## Institution('Univeristy of Utah')
    6 ## DBsubject('Algebra')
    7 ## DBchapter('Equations and Inequalities')
    8 ## DBsection('Complex Numbers')
    9 ## AuthorText1('Ron Larson and Robert P. Hostetler')
   10 ## TitleText1('Intermediate Algebra')
   11 ## EditionText1('4')
   12 ## Section1(' Radicals and Complex Numbers ')
   13 ## Problem1('')
   14 
   15 ##KEYWORDS('algebra')
   16 
   17 DOCUMENT();        # This should be the first executable line in the problem.
   18 
   19 loadMacros(
   20 "PG.pl",
   21 "PGbasicmacros.pl",
   22 "PGchoicemacros.pl",
   23 "PGanswermacros.pl",
   24 "PGauxiliaryFunctions.pl"
   25 );
   26 
   27 TEXT(&beginproblem);
   28 
   29 TEXT(EV2(<<EOT));
   30 Complete the following equation.  Your answers will be algebraic expressions.
   31 
   32 $BR
   33 \(\frac{1}{a+b i}=\) \{ ans_rule(20) \} \(+\)
   34 EOT
   35 &ANS(multivar_function_cmp("a/(a**2+b**2)",["a","b"],[[1,10],[1,10]]));
   36 TEXT(EV2(<<EOT));
   37 \{ ans_rule(20) \} \(i\)
   38 EOT
   39 &ANS(multivar_function_cmp("-b/(a**2+b**2)",["a","b"],[[1,10],[1,10]]));
   40 
   41 
   42 TEXT(EV3(<<'EOT'));
   43 $BR Hint:  This is just a special case of one of the formulas entered earlier.
   44 EOT
   45 
   46 SOLUTION(EV3(<<'EOT'));
   47 $BR $BBOLD Solution:$EBOLD  We just multiply with the conjugate complex of the denominator in numerator and denominator:
   48 \[\frac{a}{a+bi} = \frac{a-bi}{(a+bi)(a-bi)}= \frac{a-bi}{a^2+b^2} =
   49 \frac{a}{a^2+b^2} -\frac{b}{a^2+b^2}i.\]
   50 EOT
   51 
   52 
   53 ENDDOCUMENT();        # This should be the last executable line in the problem.