document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=144574>Question 144574</A>: <I><SPAN STYLE=\"word-break: break-all;\"> i need to use the discriminant to determine the number of solutions of this quadratic equation, and whether the solutions are real or complex.  i have to find the roots, but i do have to determine the number and types of solutions.  x2 + 6x - 7 = 0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #105320 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=105320\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D=b%5E2-4ac ALIGN=MIDDLE  ALT=\"D=b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the discriminant formula\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D=%286%29%5E2-4%2A1%2A-7 ALIGN=MIDDLE  ALT=\"D=%286%29%5E2-4%2A1%2A-7\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in a=1, b=6, c=-7\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D=36-4%2A1%2A-7 ALIGN=MIDDLE  ALT=\"D=36-4%2A1%2A-7\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square 6 to get 36\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D=36%2B28 ALIGN=MIDDLE  ALT=\"D=36%2B28\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply -4*1*-7 to get 28\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D=64 ALIGN=MIDDLE  ALT=\"D=64\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine 36 and 28 to get 64\r<BR>\n" );
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document.write( "Since the discriminant equals 64  (which is greater than zero) , this means  there are two real solutions. \r<BR>\n" );
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document.write( "Let's use the quadratic formula to solve for x:\r<BR>\n" );
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document.write( "Starting with the general quadratic\r<BR>\n" );
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document.write( "the general solution using the quadratic equation is:\r<BR>\n" );
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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;\" >\r<BR>\n" );
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document.write( "So lets solve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B6%2Ax-7=0 ALIGN=MIDDLE  ALT=\"x%5E2%2B6%2Ax-7=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ( notice <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=6 ALIGN=MIDDLE  ALT=\"b=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=-7 ALIGN=MIDDLE  ALT=\"c=-7\"   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=x+=+%28-6+%2B-+sqrt%28+%286%29%5E2-4%2A1%2A-7+%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"x+=+%28-6+%2B-+sqrt%28+%286%29%5E2-4%2A1%2A-7+%29%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in a=1, b=6, and c=-7\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-6+%2B-+sqrt%28+36-4%2A1%2A-7+%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"x+=+%28-6+%2B-+sqrt%28+36-4%2A1%2A-7+%29%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square 6 to get 36  \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-6+%2B-+sqrt%28+36%2B28+%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"x+=+%28-6+%2B-+sqrt%28+36%2B28+%29%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2A-7%2A1 ALIGN=MIDDLE  ALT=\"-4%2A-7%2A1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=28 ALIGN=MIDDLE  ALT=\"28\"   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=x+=+%28-6+%2B-+sqrt%28+64+%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"x+=+%28-6+%2B-+sqrt%28+64+%29%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms in the radicand (everything under the square root)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-6+%2B-+8%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"x+=+%28-6+%2B-+8%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify the square root (note: If you need help with simplifying the square root, check out this <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Radicals/simplifying-square-roots.solver> solver</a>)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-6+%2B-+8%29%2F2 ALIGN=MIDDLE  ALT=\"x+=+%28-6+%2B-+8%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply 2 and 1 to get 2\r<BR>\n" );
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document.write( "So now the expression breaks down into two parts\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-6+%2B+8%29%2F2 ALIGN=MIDDLE  ALT=\"x+=+%28-6+%2B+8%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-6+-+8%29%2F2 ALIGN=MIDDLE  ALT=\"x+=+%28-6+-+8%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Lets look at the first part:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2%2F2 ALIGN=MIDDLE  ALT=\"x=2%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add the terms in the numerator\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide\r<BR>\n" );
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document.write( "So one answer is\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now lets look at the second part:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-6+-+8%29%2F2 ALIGN=MIDDLE  ALT=\"x=%28-6+-+8%29%2F2\"   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=x=-14%2F2 ALIGN=MIDDLE  ALT=\"x=-14%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract the terms in the numerator\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-7 ALIGN=MIDDLE  ALT=\"x=-7\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide\r<BR>\n" );
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document.write( "So another answer is\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-7 ALIGN=MIDDLE  ALT=\"x=-7\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So our solutions are:\r<BR>\n" );
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document.write( " </SPAN>\n" );
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