document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=251829>Question 251829</A>: <I><SPAN STYLE=\"word-break: break-all;\"> x^2+2x+9\r<BR>\n" );
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document.write( "I have tried x={-2ħsqrt(2^2-4*1*9)}/2*1  <BR>\n" );
document.write( "(quadratic formula: x={-bħsqrt( b^2-4ac)}/2(a))<BR>\n" );
document.write( "Process:<BR>\n" );
document.write( "x={-2ħsqrt(2^2-4*1*9)}/2(1)<BR>\n" );
document.write( "x={-2ħsqrt(4-36)}/2<BR>\n" );
document.write( "x=(-2/2)ħ{sqrt-32/2}<BR>\n" );
document.write( "x=-1ħ(sqrt-32/2)\r<BR>\n" );
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document.write( "Looking for -1ħ2i*sqrt(2), but I can't get there. <BR>\n" );
document.write( "I need the process to find this imaginary root.<BR>\n" );
document.write( "what am I doing wrong? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #183524 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=183524\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The trick here lies in simplifying <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28-32%29 ALIGN=MIDDLE  ALT=\"sqrt%28-32%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28-32%29 ALIGN=MIDDLE  ALT=\"sqrt%28-32%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given expression \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28-1%2A32%29 ALIGN=MIDDLE  ALT=\"sqrt%28-1%2A32%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out a negative 1\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28-1%29%2Asqrt%2832%29 ALIGN=MIDDLE  ALT=\"sqrt%28-1%29%2Asqrt%2832%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the square roots using the identity <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%2Ay%29=sqrt%28x%29%2Asqrt%28y%29 ALIGN=MIDDLE  ALT=\"sqrt%28x%2Ay%29=sqrt%28x%29%2Asqrt%28y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i%2Asqrt%2832%29 ALIGN=MIDDLE  ALT=\"i%2Asqrt%2832%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28-1%29 ALIGN=MIDDLE  ALT=\"sqrt%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i ALIGN=MIDDLE  ALT=\"i\"   DISABLED_style= \"max-width:90%; height:auto;\" > (remember <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i=sqrt%28-1%29 ALIGN=MIDDLE  ALT=\"i=sqrt%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "Now lets simplify <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2832%29 ALIGN=MIDDLE  ALT=\"sqrt%2832%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
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document.write( "The goal of simplifying expressions with square roots is to factor the radicand into a product of two numbers. One of these two numbers must be a perfect square. When you take the square root of this perfect square, you will get a rational number.\r<BR>\n" );
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document.write( "So let's list the factors of 32\r<BR>\n" );
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document.write( "Factors:\r<BR>\n" );
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document.write( "1, 2, 4, 8, 16, 32\r<BR>\n" );
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document.write( "Notice how 16 is the largest perfect square, so lets factor 32 into 16*2\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2816%2A2%29 ALIGN=MIDDLE  ALT=\"sqrt%2816%2A2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor 32 into 16*2<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2816%29%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"sqrt%2816%29%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the square roots using the identity <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%2Ay%29=sqrt%28x%29%2Asqrt%28y%29 ALIGN=MIDDLE  ALT=\"sqrt%28x%2Ay%29=sqrt%28x%29%2Asqrt%28y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"4%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the square root of the perfect square 16 to get 4 <BR>\n" );
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document.write( "This means that the expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2832%29 ALIGN=MIDDLE  ALT=\"sqrt%2832%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > simplifies to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"4%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28-32%29 ALIGN=MIDDLE  ALT=\"sqrt%28-32%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > simplifies to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Ai%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"4%2Ai%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (just reintroduce <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i ALIGN=MIDDLE  ALT=\"i\"   DISABLED_style= \"max-width:90%; height:auto;\" > back in)\r<BR>\n" );
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document.write( "In other words, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28-32%29=4%2Ai%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"sqrt%28-32%29=4%2Ai%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Going back to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-2%2B-sqrt%28-32%29%29%2F2 ALIGN=MIDDLE  ALT=\"x=%28-2%2B-sqrt%28-32%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this expression simplifies to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-2%2B-+4i%2Asqrt%282%29%29%2F2 ALIGN=MIDDLE  ALT=\"x=%28-2%2B-+4i%2Asqrt%282%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. From there, it reduces to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1%2B-+2i%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"x=-1%2B-+2i%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the solutions are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1%2B+2i%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"x=-1%2B+2i%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1-+2i%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"x=-1-+2i%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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