document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=41299>Question 41299</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0321233832/algebrahomeworkh></A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> Help<BR>\n" );
document.write( "What type of solution do you get for quadratic equations where D < 0? Give reasons for your answer. Also provide an example of such a quadratic equation. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #26671 by </B><A HREF=/tutors/aboutme.mpl?userid=psbhowmick><B>psbhowmick(878)</B></A><img src=\"/images/stars/stars-11.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=psbhowmick><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=psbhowmick><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=26671\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> By D you meant determinant. Right?\r<BR>\n" );
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document.write( "For a quadratic equation with real coefficients, if D < 0 then the roots are imaginery and occur in conjugate pairs as well.<BR>\n" );
document.write( "Thus if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2+%2B+bx+%2B+c+=+0 ALIGN=MIDDLE  ALT=\"ax%5E2+%2B+bx+%2B+c+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > be a quadratic equation where a,b,c are real then if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4%2Aa%2Ac+%3C0 ALIGN=MIDDLE  ALT=\"b%5E2-4%2Aa%2Ac+%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > then the roots are of the form A + iB and A - iB where A and B are real.\r<BR>\n" );
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document.write( "Example: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%5E2+%2B+8x+%2B+5+=+0 ALIGN=MIDDLE  ALT=\"5x%5E2+%2B+8x+%2B+5+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a quadratic equation with real coefficients.<BR>\n" );
document.write( "The determinant is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%5E2+-+4%2A5%2A5 ALIGN=MIDDLE  ALT=\"8%5E2+-+4%2A5%2A5\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 64 - 100 = -36 < 0.<BR>\n" );
document.write( "Now let us find the roots.<BR>\n" );
document.write( "Sreedharachary's formula is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-8+%2B-+sqrt%28+-36+%29%29%2F10+ ALIGN=MIDDLE  ALT=\"x+=+%28-8+%2B-+sqrt%28+-36+%29%29%2F10+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-8+%2B-+sqrt%28+%286%2Ai%29%5E2+%29%29%2F10+ ALIGN=MIDDLE  ALT=\"x+=+%28-8+%2B-+sqrt%28+%286%2Ai%29%5E2+%29%29%2F10+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-8+%2B-+6%2Ai%29%2F10+ ALIGN=MIDDLE  ALT=\"x+=+%28-8+%2B-+6%2Ai%29%2F10+\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-0.8+%2B-+0.6%2Ai+ ALIGN=MIDDLE  ALT=\"-0.8+%2B-+0.6%2Ai+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Thus either x = -0.8 + 0.6i or x = -0.8 - 0.6i\r<BR>\n" );
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document.write( "So you can see that the roots are complex conjugate. </SPAN>\n" );
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