document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210618>Question 1210618</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Fill in the blanks to make the equation true.\r<BR>\n" );
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document.write( "15a^4 - 5a^4 b - ___ + 20a^2 = 12a^2 b - 8b^4 - 4^b + ___\r<BR>\n" );
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document.write( "4a^2 b^2 12ab^2 18a^2 5a^2 b^3 16b^2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854433 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5362)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=854433\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This question is strange, and open to several interpretations.<BR>\n" );
document.write( "I also suspect that the question was not properly typed as intended.<BR>\n" );
document.write( "Among other problems, the term 4^b<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%5Eb ALIGN=MIDDLE  ALT=\"4%5Eb\"   DISABLED_style= \"max-width:90%; height:auto;\" > in the equation seems suspicious.<BR>\n" );
document.write( "I tried several interpretations and I cannot find a valid solution.<BR>\n" );
document.write( "Maybe equation was meant to be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15a%5E4+-+5a%5E4%2Ab+-+%22_____%22+%2B+20a%5E2+=+12a%5E2%2Ab+-+8b%5E4+-+4%5Eb+%2B+%22_____%22 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Or maybe it was meant to be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15a%5E4+-+5a%5E4b+-+%22_____%22+%2B+20a%5E2+=+12a%5E2b+-+8b%5E4+-+4%5Eb+%2B+%22_____%22 ALIGN=MIDDLE  ALT=\"15a%5E4+-+5a%5E4b+-+%22_____%22+%2B+20a%5E2+=+12a%5E2b+-+8b%5E4+-+4%5Eb+%2B+%22_____%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Or maybe none of the above. The term <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%5Eb ALIGN=MIDDLE  ALT=\"4%5Eb\"   DISABLED_style= \"max-width:90%; height:auto;\" >} looks suspiciously out of place.<BR>\n" );
document.write( "and what was meant by \"make the equation true\" was<BR>\n" );
document.write( "make the equation true for any ordered pair (a,b).<BR>\n" );
document.write( "The expressions to be used to fill the blanks seem to be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%281%2C13%2C4a%5E2%2C%22+%2C+%22%2Cb%5E2%2C%22+%2C+%22%2C12ab%5E2%2C%22+%2C+%22%2C18a%5E2%2C%22+%2C+%22%2C5a%5E2%2C%22+%2C+%22%2Cb%5E3%2C%22+%2C+%22%2C16b%5E2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "or maybe <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%281%2C9%2C4a%5E2b%5E2%2C%22+%2C+%22%2C12ab%5E2%2C%22+%2C+%22%2C18a%5E2%2C%22+%2C+%22%2C5a%5E2b%5E3%2C%22+%2C+%22%2C16b%5E2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "What was meant by what was meant by \"make the equation true\" is not clear, or a commonly stated that way in math.<BR>\n" );
document.write( "Maybe equation was meant to be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15a%5E4+-+5a%5E4%2Ab+-+%22_____%22+%2B+20a%5E2+=+12a%5E2%2Ab+-+8b%5E4+-+4%5Eb+%2B+%22_____%22 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Or maybe it was meant to be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15a%5E4+-+5a%5E4b+-+%22_____%22+%2B+20a%5E2+=+12a%5E2b+-+8b%5E4+-+4%5Eb+%2B+%22_____%22 ALIGN=MIDDLE  ALT=\"15a%5E4+-+5a%5E4b+-+%22_____%22+%2B+20a%5E2+=+12a%5E2b+-+8b%5E4+-+4%5Eb+%2B+%22_____%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Did it mean make the equation be an identity, true for any ordered pair (a,b)?<BR>\n" );
document.write( "Are we to assume that \"fill in the blanks\", with a list of expressions at the end means to use one of the expressions given to fill one of the blanks and another one of the expressions given to fill the other blank?<BR>\n" );
document.write( " <BR>\n" );
document.write( "Making the two assumptions above, for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28a=0%2Cb=1%29 ALIGN=MIDDLE  ALT=\"system%28a=0%2Cb=1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%281%2C13%2C4a%5E2%2C%22+%2C+%22%2Cb%5E2%2C%22+%2C+%22%2C12ab%5E2%2C%22+%2C+%22%2C18a%5E2%2C%22+%2C+%22%2C5a%5E2%2C%22+%2C+%22%2Cb%5E3%2C%22+%2C+%22%2C16b%5E2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > , turns into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%281%2C13%2C0%2C%22+%2C+%22%2C1%2C%22+%2C+%22%2C0%2C%22+%2C+%22%2C0%2C%22+%2C+%22%2C0%2C%22+%2C+%22%2C1%2C%22+%2C+%22%2C16%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "while <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%281%2C9%2C4a%5E2b%5E2%2C%22+%2C+%22%2C12ab%5E2%2C%22+%2C+%22%2C18a%5E2%2C%22+%2C+%22%2C5a%5E2b%5E3%2C%22+%2C+%22%2C16b%5E2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > turns into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%281%2C9%2C0%2C%22+%2C+%22%2C0%2C%22+%2C+%22%2C0%2C%22+%2C+%22%2C0%2C%22+%2C+%22%2C16%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28a=0%2Cb=1%29 ALIGN=MIDDLE  ALT=\"system%28a=0%2Cb=1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15a%5E4+-+5a%5E4%2Ab+-+%22_____%22+%2B+20a%5E2+=+12a%5E2%2Ab+-+8b%5E4+-+4%5Eb+%2B+%22_____%22 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "which I will re-write as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15a%5E4+-+5a%5E4%2Ab+-+blank%5B1%5D+%2B+20a%5E2+=+12a%5E2%2Ab+-+8b%5E4+-+4%5Eb+%2B+blank%5B2%5D ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > turns into<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+-+0-+blank%5B1%5D+%2B+0+=+0+-+8%2A1%5E4+-+4%5E1+%2B+blank%5B2%5D ALIGN=MIDDLE  ALT=\"0+-+0-+blank%5B1%5D+%2B+0+=+0+-+8%2A1%5E4+-+4%5E1+%2B+blank%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > , which simplifies into<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-+blank%5B1%5D=+-+8-+4+%2B+blank%5B2%5D ALIGN=MIDDLE  ALT=\"-+blank%5B1%5D=+-+8-+4+%2B+blank%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-+blank%5B1%5D=-12%2Bblank%5B2%5D ALIGN=MIDDLE  ALT=\"-+blank%5B1%5D=-12%2Bblank%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blank%5B1%5D%2Bblank%5B2%5D=12 ALIGN=MIDDLE  ALT=\"blank%5B1%5D%2Bblank%5B2%5D=12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28a=0%2Cb=1%29 ALIGN=MIDDLE  ALT=\"system%28a=0%2Cb=1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15a%5E4+-+5a%5E4b+-+%22_____%22+%2B+20a%5E2+=+12a%5E2b+-+8b%5E4+-+4%5Eb+%2B+%22_____%22 ALIGN=MIDDLE  ALT=\"15a%5E4+-+5a%5E4b+-+%22_____%22+%2B+20a%5E2+=+12a%5E2b+-+8b%5E4+-+4%5Eb+%2B+%22_____%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "which I will re-write as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15a%5E4+-+5a%5E4b+-+blank%5B1%5D%2B+20a%5E2+=+12a%5E2b+-+8b%5E4+-+4%5Eb+%2Bblank%5B2%5D ALIGN=MIDDLE  ALT=\"15a%5E4+-+5a%5E4b+-+blank%5B1%5D%2B+20a%5E2+=+12a%5E2b+-+8b%5E4+-+4%5Eb+%2Bblank%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,  turns into<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+-+0+-+blank%5B1%5D%2B+0+=+0+-+8%2A1%5E4+-+4%5E1+%2Bblank%5B2%5D ALIGN=MIDDLE  ALT=\"0+-+0+-+blank%5B1%5D%2B+0+=+0+-+8%2A1%5E4+-+4%5E1+%2Bblank%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > , which also simplifies into<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-blank%5B1%5D=-8-4%2Bblank%5B2%5D ALIGN=MIDDLE  ALT=\"-blank%5B1%5D=-8-4%2Bblank%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "in the list of values to fill the blank above, there is no 2 values that add to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12 ALIGN=MIDDLE  ALT=\"12\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
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