document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=135617>Question 135617</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Write a quadratic equation for which the following are solutions for x: ± 3i<BR>\n" );
document.write( "This one has me a bit confused. Here's what I get<BR>\n" );
document.write( "x = ±i sqrt 9<BR>\n" );
document.write( "x = ± sqrt -9<BR>\n" );
document.write( "x^2=-9<BR>\n" );
document.write( "x^2+9=0<BR>\n" );
document.write( "That's not a complete quadratic equaiton, but I don't know what to do next or if I'm doing it right to begin with. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #99355 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=99355\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You sort of took a convoluted path to get there, but you did arrive at the right answer.\r<BR>\n" );
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document.write( "Here's how I would have done it:\r<BR>\n" );
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document.write( "A polynomial equation has a root <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=a ALIGN=MIDDLE  ALT=\"x=a\"   DISABLED_style= \"max-width:90%; height:auto;\" > if and only if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-a ALIGN=MIDDLE  ALT=\"x-a\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a factor of the polynomial.  Furthermore, a polynomial equation of degree n has exactly n roots.\r<BR>\n" );
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document.write( "You are given two roots, so you know that you must have a quadratic (degree 2) polynomial equation.\r<BR>\n" );
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document.write( "The roots are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3i ALIGN=MIDDLE  ALT=\"3i\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3i ALIGN=MIDDLE  ALT=\"-3i\"   DISABLED_style= \"max-width:90%; height:auto;\" >, therefore, the factors of the polynomial must be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-3i ALIGN=MIDDLE  ALT=\"x-3i\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-%28-3i%29=x%2B3i ALIGN=MIDDLE  ALT=\"x-%28-3i%29=x%2B3i\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Multiply the two factors together:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-3i%29%28x%2B3i%29=x%5E2%2B9 ALIGN=MIDDLE  ALT=\"%28x-3i%29%28x%2B3i%29=x%5E2%2B9\"   DISABLED_style= \"max-width:90%; height:auto;\" > (factorization of the difference of two squares in reverse)\r<BR>\n" );
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document.write( "So your equation is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B9=0 ALIGN=MIDDLE  ALT=\"x%5E2%2B9=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "And, by the way, it most certainly is a 'complete' quadratic equation of the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=1 ALIGN=MIDDLE  ALT=\"a=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=0 ALIGN=MIDDLE  ALT=\"b=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=9 ALIGN=MIDDLE  ALT=\"c=9\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  The only restriction on the possible values for the coefficients in  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > is that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%3C%3E0 ALIGN=MIDDLE  ALT=\"a%3C%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  b and c can be any values, including zero. </SPAN>\n" );
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