document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1205480>Question 1205480</A>: <I><SPAN STYLE=\"word-break: break-all;\"> if a + b = -3 and b - c = 6, find the value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a%5E2+-+3b%5E2+%2B+c%5E2 ALIGN=MIDDLE  ALT=\"2a%5E2+-+3b%5E2+%2B+c%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #842317 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13203)</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=842317\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Note that the problem does not ask us to find the values of a, b, and c.<br><BR>\n" );
document.write( "In fact there are an infinite number of triples of numbers a, b, and c which all give the same value for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a%5E2+-+3b%5E2+%2B+c%5E2 ALIGN=MIDDLE  ALT=\"2a%5E2+-+3b%5E2+%2B+c%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  This is easy to see empirically by choosing arbitrary values for a and finding the corresponding values of b and c using the given equations; in every case the value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a%5E2+-+3b%5E2+%2B+c%5E2 ALIGN=MIDDLE  ALT=\"2a%5E2+-+3b%5E2+%2B+c%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the same.<br><BR>\n" );
document.write( "ANSWER: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a%5E2+-+3b%5E2+%2B+c%5E2+=+54 ALIGN=MIDDLE  ALT=\"2a%5E2+-+3b%5E2+%2B+c%5E2+=+54\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "Let's use algebra to show that 54 is always the answer.<br><BR>\n" );
document.write( "One common way to solve problems like this is to square the given equations.  But that introduces \"ab\" and \"bc\" terms, which we really don't want.<br><BR>\n" );
document.write( "So another way to solve the problem is to look for examples of expressions of the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-y%5E2=%28x%2By%29%28x-y%29 ALIGN=MIDDLE  ALT=\"x%5E2-y%5E2=%28x%2By%29%28x-y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > in the given expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a%5E2+-+3b%5E2+%2B+c%5E2 ALIGN=MIDDLE  ALT=\"2a%5E2+-+3b%5E2+%2B+c%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a%5E2+-+3b%5E2+%2B+c%5E2=2a%5E2-2b%5E2-b%5E2%2Bc%5E2=2%28a%5E2-b%5E2%29-%28b%5E2-c%5E2%29=2%28a-b%29%28a%2Bb%29-%28b%2Bc%29%28b-c%29=2%28a-b%29%28-3%29-%28b%2Bc%29%286%29=-6a%2B6b-6b-6c=-6%28a%2Bc%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "But eliminating b from the original two equations gives us <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bc=-9 ALIGN=MIDDLE  ALT=\"a%2Bc=-9\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  And so<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a%5E2-3b%5E2%2Bc%5E2=%28-6%29%28-9%29=54 ALIGN=MIDDLE  ALT=\"2a%5E2-3b%5E2%2Bc%5E2=%28-6%29%28-9%29=54\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "ANSWER: 54<br><BR>\n" );
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