document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=515414>Question 515414</A>: <I><SPAN STYLE=\"word-break: break-all;\"> factor completely 112b^2+392bg+343g^2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #343956 by </B><A HREF=/tutors/aboutme.mpl?userid=drcole><B>drcole(72)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=drcole><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=343956\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=112+b%5E2+%2B+392+b+g+%2B+343+g%5E2+ ALIGN=MIDDLE  ALT=\"112+b%5E2+%2B+392+b+g+%2B+343+g%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we first notice that all of the coefficients are divisible by 7, so we factor 7 out of each term:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+7%2816+b%5E2+%2B+56++b+g+%2B+49+g%5E2%29+ ALIGN=MIDDLE  ALT=\"+7%2816+b%5E2+%2B+56++b+g+%2B+49+g%5E2%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "There are no other common factors of the coefficients.  Now we need to find two numbers whose product is (16)(49) = 784 and whose sum is 56.  Let's write out some possible ways to factor 784 and look at the sum of the factors:\r<BR>\n" );
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document.write( "(1)(784) sum = 785<BR>\n" );
document.write( "(2)(392) sum = 394<BR>\n" );
document.write( "(4)(196) sum = 200<BR>\n" );
document.write( "(7)(112) sum = 119<BR>\n" );
document.write( "(8)(98) sum = 106<BR>\n" );
document.write( "(14)(56) sum = 70<BR>\n" );
document.write( "(16)(49) sum = 65<BR>\n" );
document.write( "(28)(28) sum = 56 <--- this works!\r<BR>\n" );
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document.write( "So we've found our combination.  We can rewrite the quadratic expression using these two factors:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2816+b%5E2+%2B+28+b+g+%2B+28+b+g+%2B+343+g%5E2%29 ALIGN=MIDDLE  ALT=\"7%2816+b%5E2+%2B+28+b+g+%2B+28+b+g+%2B+343+g%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now we can factor the quadratic by grouping.  The greatest common factor of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16+b%5E2 ALIGN=MIDDLE  ALT=\"16+b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=28+b+g ALIGN=MIDDLE  ALT=\"28+b+g\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4b ALIGN=MIDDLE  ALT=\"4b\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so we factor that out of the first two terms:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%284b%28+4b+%2B+7g%29+%2B+28+b+g+%2B+343+g%5E2%29 ALIGN=MIDDLE  ALT=\"7%284b%28+4b+%2B+7g%29+%2B+28+b+g+%2B+343+g%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The greatest common factor of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=28+b+g ALIGN=MIDDLE  ALT=\"28+b+g\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=343+g%5E2 ALIGN=MIDDLE  ALT=\"343+g%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7g ALIGN=MIDDLE  ALT=\"7g\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so we factor this out of the last two terms:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%284b%284b+%2B+7g%29+%2B+7g%284b+%2B+7g%29%29 ALIGN=MIDDLE  ALT=\"7%284b%284b+%2B+7g%29+%2B+7g%284b+%2B+7g%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now we factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4b+%2B+7g ALIGN=MIDDLE  ALT=\"4b+%2B+7g\"   DISABLED_style= \"max-width:90%; height:auto;\" > out of both terms:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%284b+%2B+7g%29%284b+%2B+7g%29+=+7%284b+%2B+7g%29%5E2 ALIGN=MIDDLE  ALT=\"7%284b+%2B+7g%29%284b+%2B+7g%29+=+7%284b+%2B+7g%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "This completes the factorization. </SPAN>\n" );
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