document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1148967>Question 1148967</A>: <I><SPAN STYLE=\"word-break: break-all;\"> solve  : (x² + 3x +1)^(x² +7x +12) = 1  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #770303 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</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=770303\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "The equation is satisfied if...<br><BR>\n" );
document.write( "(1) (x^2+7x+12) = 0  (any number to the 0 power is 1)<br><BR>\n" );
document.write( "(2) (x^2 +3x+1) = 1  (1 raised to any power is 1)<br><BR>\n" );
document.write( "(3) (x^2+3x+1) = -1 AND (x^2+7x+12) is even  (-1 raised to an even power is 1)<br><BR>\n" );
document.write( "(1) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B7x%2B12+=+%28x%2B3%29%28x%2B4%29+=+0 ALIGN=MIDDLE  ALT=\"x%5E2%2B7x%2B12+=+%28x%2B3%29%28x%2B4%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "x = -3 and x = -4 are solutions.<br><BR>\n" );
document.write( "(2) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B3x%2B1+=+1 ALIGN=MIDDLE  ALT=\"x%5E2%2B3x%2B1+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B3x+=+0 ALIGN=MIDDLE  ALT=\"x%5E2%2B3x+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%28x%2B3%29+=+0 ALIGN=MIDDLE  ALT=\"x%28x%2B3%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "x = -3 and x = 0 are solutions (x = -3 is already known to be a solution).<br><BR>\n" );
document.write( "(3) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B3x%2B1+=+-1 ALIGN=MIDDLE  ALT=\"x%5E2%2B3x%2B1+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B3x%2B2+=+0 ALIGN=MIDDLE  ALT=\"x%5E2%2B3x%2B2+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B1%29%28x%2B2%29+=+0 ALIGN=MIDDLE  ALT=\"%28x%2B1%29%28x%2B2%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "x = -1 and x = -2 are solutions IF they make x^2+7x+12 = (x+3)(x+4) even.<br><BR>\n" );
document.write( "Note that, for both x = -1 and x = -2, (x+3)(x+4) are consecutive integers.  One of them has to be even and one odd; so their product is even.<br<<BR>\n" );
document.write( "Therefore, both x = -1 and x = -2 are solutions.<br><BR>\n" );
document.write( "ANSWER: The solutions are 0, -1 -2, -3, and -4.<br><BR>\n" );
document.write( " </SPAN>\n" );
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