document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1115736>Question 1115736</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 	<BR>\n" );
document.write( "Find the value of c so that the equation will have exactly one rational solution.<BR>\n" );
document.write( "p^2-10p+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 #730562 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=730562\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=\"black\" face=\"times\" size=\"3\"><BR>\n" );
document.write( "First I'm going to alter <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%5E2-10p%2Bc=0 ALIGN=MIDDLE  ALT=\"p%5E2-10p%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > so that it's in a more familiar form<BR>\n" );
document.write( "p^2-10p+c=0<BR>\n" );
document.write( "x^2-10x+c=0 ... replace every 'p' with 'x'<BR>\n" );
document.write( "1x^2-10x+c=0 ... rewrite x^2 as 1x^2<BR>\n" );
document.write( "1x^2+(-10)x+c=0 ... rewrite the \"1x^2-10x\" as \"1x^2+(-10)x\"\r<BR>\n" );
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document.write( "Note how <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%5E2%2B%28-10%29x%2Bc=0 ALIGN=MIDDLE  ALT=\"1x%5E2%2B%28-10%29x%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > is in the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "It might be better to line up the terms like so<BR>\n" );
document.write( "1x^2+(-10)x+c=0<BR>\n" );
document.write( "ax^2+bx+c=0\r<BR>\n" );
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document.write( "Which helps us see that <BR>\n" );
document.write( "a = 1<BR>\n" );
document.write( "b = -10<BR>\n" );
document.write( "c = unknown for now\r<BR>\n" );
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document.write( "Now turn to the discriminant formula<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;\" ><BR>\n" );
document.write( "Exactly one rational solution occurs when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+0 ALIGN=MIDDLE  ALT=\"d+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So we'll plug in a = 1, b = -10, and d = 0. Then we'll solve for c<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;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=+%28-10%29%5E2+-+4%281%29c ALIGN=MIDDLE  ALT=\"0+=+%28-10%29%5E2+-+4%281%29c\"   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=0+=+100-4c ALIGN=MIDDLE  ALT=\"0+=+100-4c\"   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=4c+=+100 ALIGN=MIDDLE  ALT=\"4c+=+100\"   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=c+=+100%2F4 ALIGN=MIDDLE  ALT=\"c+=+100%2F4\"   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=c+=+25 ALIGN=MIDDLE  ALT=\"c+=+25\"   DISABLED_style= \"max-width:90%; height:auto;\" > (this is the final answer)\r<BR>\n" );
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document.write( "So the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%5E2-10p%2Bc=0 ALIGN=MIDDLE  ALT=\"p%5E2-10p%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > becomes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%5E2-10p%2B25=0 ALIGN=MIDDLE  ALT=\"p%5E2-10p%2B25=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "I'll leave it as an exercise for you to confirm that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%5E2-10p%2B25=0 ALIGN=MIDDLE  ALT=\"p%5E2-10p%2B25=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > has exactly one solution, and to find the solution. Though this is optional, its good to practice. Hint: Quadratic Formula<BR>\n" );
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