document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=116197>Question 116197</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How do you find if an equation has 2 real number solutions, 1 real solution or no real-number solutuons? an example is      x^2-6x+c=0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #84504 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=84504\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> From the quadratic formula\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( "the discriminant consists of all of the terms in the square root. So the discriminant is\r<BR>\n" );
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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;\" >\r<BR>\n" );
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document.write( "the discriminant tells us how many solutions (and what type of solutions) we can expect for any quadratic.\r<BR>\n" );
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document.write( "So let's find the discriminant for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-6x%2Bc ALIGN=MIDDLE  ALT=\"x%5E2-6x%2Bc\"   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=D=-6%5E2-4%2A1%2Ac ALIGN=MIDDLE  ALT=\"D=-6%5E2-4%2A1%2Ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in a=1, b=-6\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%2Ac ALIGN=MIDDLE  ALT=\"D=36-4%2A1%2Ac\"   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=36-4c ALIGN=MIDDLE  ALT=\"D=36-4c\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply -4*1*c to get -4c\r<BR>\n" );
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document.write( "Now if the discriminant is greater than zero, then we'll have 2 real solutions.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=36-4c%3E0 ALIGN=MIDDLE  ALT=\"36-4c%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Set the discriminant greater than zero\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4c%3E0-36 ALIGN=MIDDLE  ALT=\"-4c%3E0-36\"   DISABLED_style= \"max-width:90%; height:auto;\" >Subtract 36 from both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4c%3E-36 ALIGN=MIDDLE  ALT=\"-4c%3E-36\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms on the right side\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%3C%28-36%29%2F%28-4%29 ALIGN=MIDDLE  ALT=\"c%3C%28-36%29%2F%28-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by -4 to isolate c  (note: Remember, dividing both sides by a negative number flips the inequality sign) \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%3C9 ALIGN=MIDDLE  ALT=\"c%3C9\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide\r<BR>\n" );
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document.write( "So when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%3C9 ALIGN=MIDDLE  ALT=\"c%3C9\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we'll have two real solutions.\r<BR>\n" );
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document.write( "Now when the discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=36-4c ALIGN=MIDDLE  ALT=\"36-4c\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equal to zero, then we'll have one real solution:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=36-4c=0 ALIGN=MIDDLE  ALT=\"36-4c=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Set the discriminant equal to zero\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4c=0-36 ALIGN=MIDDLE  ALT=\"-4c=0-36\"   DISABLED_style= \"max-width:90%; height:auto;\" >Subtract 36 from both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4c=-36 ALIGN=MIDDLE  ALT=\"-4c=-36\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms on the right side\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=%28-36%29%2F%28-4%29 ALIGN=MIDDLE  ALT=\"c=%28-36%29%2F%28-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by -4 to isolate c\r<BR>\n" );
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document.write( "<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;\" > Divide\r<BR>\n" );
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document.write( "So when <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;\" >, we'll have one real solution.\r<BR>\n" );
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document.write( "Now when the discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=36-4c ALIGN=MIDDLE  ALT=\"36-4c\"   DISABLED_style= \"max-width:90%; height:auto;\" > is less than zero, then we'll have no real solutions:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=36-4c%3C0 ALIGN=MIDDLE  ALT=\"36-4c%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Set the discriminant less than zero\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4c%3C0-36 ALIGN=MIDDLE  ALT=\"-4c%3C0-36\"   DISABLED_style= \"max-width:90%; height:auto;\" >Subtract 36 from both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4c%3C-36 ALIGN=MIDDLE  ALT=\"-4c%3C-36\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms on the right side\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%3E%28-36%29%2F%28-4%29 ALIGN=MIDDLE  ALT=\"c%3E%28-36%29%2F%28-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by -4 to isolate c  (note: Remember, dividing both sides by a negative number flips the inequality sign) \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%3E9 ALIGN=MIDDLE  ALT=\"c%3E9\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide\r<BR>\n" );
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document.write( "So when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%3E9 ALIGN=MIDDLE  ALT=\"c%3E9\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we won't have any real solutions </SPAN>\n" );
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