document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=235130>Question 235130</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solving Quadratic Equations by graphing; there is either 1, 2, or no real solutions... can you help me with this?\r<BR>\n" );
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document.write( "x^2-2x-24=0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #173309 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=173309\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "First you want to graph the function <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE f(x)\ =\ y\ =\ x^2\ -\ 2x\ -\ 24\">\r<BR>\n" );
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document.write( "Then you want to find the one place that the graph of the function is tangent to the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">-axis, meaning that you have 1 real solution (actually, you would say that the equation has one real root with a multiplicity of two), or you want to find the two places that the graph intersects the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">-axis, meaning that you have two distinct real solutions, or you want to discover that the graph never touches the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">-axis, meaning that the equation has no real solutions.\r<BR>\n" );
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document.write( "What you are doing is answering the question, \"What number or numbers can I substitute for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> so that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large f(x)\ =\ 0\">?\"  The function has a value of zero at the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">-axis.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28%0D%0A400%2C+600%2C+-10%2C+10%2C-30%2C+30%2C%0D%0Agrid%281%29%2C%0D%0Agraph%28%0D%0A400%2C+600%2C+-10%2C+10%2C-30%2C+30%2C%0D%0Ax%5E2-2x-24%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Compare this to: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE f(x)\ =\ y\ =\ x^2\ -\ 2x\ +\ 1\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28%0D%0A400%2C+600%2C+-10%2C+10%2C-30%2C+30%2C%0D%0Agrid%281%29%2C%0D%0Agraph%28%0D%0A400%2C+600%2C+-10%2C+10%2C-30%2C+30%2C%0D%0Ax%5E2-2x%2B1%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Notice two things:  1.  The function is a perfect square trinomial.  2.  The graph is tangent to the axis.  One root with a multiplicity of 2.\r<BR>\n" );
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document.write( "No compare to: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE f(x)\ =\ y\ =\ x^2\ -\ 2x\ +\ 4\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28%0D%0A400%2C+600%2C+-10%2C+10%2C-30%2C+30%2C%0D%0Agrid%281%29%2C%0D%0Agraph%28%0D%0A400%2C+600%2C+-10%2C+10%2C-30%2C+30%2C%0D%0Ax%5E2-2x%2B4%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Notice that the graph does not touch the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">-axis at all.  No real solutions.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
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