document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210612>Question 1210612</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Fill in the blanks to make an equation that is linear in disguise, and has a solution.<BR>\n" );
document.write( "sqrt(2x - 3) + __x + ___ = ___x - ___ + ____<BR>\n" );
document.write( "8 9 10 11 12 13 14 15 16 17 18 19 20  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854413 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13351)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=854413\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "It is not at all clear what the purpose of this problem is.  You can find numerous solutions either by pure trial and error or using formal algebra.<br><BR>\n" );
document.write( "And, as another tutor pointed out, it is not clear whether numbers from the given list can be used more than once.<br><BR>\n" );
document.write( "Using A, B, C, D, and E to represent the numbers in the blanks, the given equation is<br><BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282x-3%29%2BAx%2BB=Cx-D%2BE ALIGN=MIDDLE  ALT=\"sqrt%282x-3%29%2BAx%2BB=Cx-D%2BE\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "This equation can be \"linear in disguise\" if x is any number for which <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-3 ALIGN=MIDDLE  ALT=\"2x-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a perfect square.<br><BR>\n" );
document.write( "Even limiting ourselves to integer perfect squares, there are an infinite number of possible values for x:<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
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document.write( "    2x-3   x\r\n" );
document.write( "   ----------\r\n" );
document.write( "     0    3/2\r\n" );
document.write( "     1      2\r\n" );
document.write( "     9      6\r\n" );
document.write( "    25     14\r\n" );
document.write( "    49     26\r\n" );
document.write( "    ...</PRE><BR>\n" );
document.write( "For many of those values of x the equation will have solutions (often multiple solutions) with A, B, C, D, and E being values from the given list.<br><BR>\n" );
document.write( "But <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-3 ALIGN=MIDDLE  ALT=\"2x-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be a perfect square which is not an integer; that will lead us to a whole new family of possible values for x, and possibly many of those will allow additional solutions to the problem.<br><BR>\n" );
document.write( "Finally <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-3 ALIGN=MIDDLE  ALT=\"2x-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > does not have to be a perfect square.  We can isolate the radical and square both sides of the equation to get yet another whole family of possible solutions to the problem.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282x-3%29=Cx-D%2BE-Ax-B ALIGN=MIDDLE  ALT=\"sqrt%282x-3%29=Cx-D%2BE-Ax-B\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282x-3%29=%28C-A%29x%2B%28-D%2BE-B%29 ALIGN=MIDDLE  ALT=\"sqrt%282x-3%29=%28C-A%29x%2B%28-D%2BE-B%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Squaring both sides will yield a quadratic equation; since we are supposed to get a linear equation, it would be necessary for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C-A ALIGN=MIDDLE  ALT=\"C-A\"   DISABLED_style= \"max-width:90%; height:auto;\" > to be zero -- i.e., A and C would have to be the same value from the given list.  If that is indeed allowed, then we open up the possibility of many more solutions to the problem.<br>\r<BR>\n" );
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