document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=180022>Question 180022</A>: <I><SPAN STYLE=\"word-break: break-all;\"> a recent homework example said fine the domain of the function:<BR>\n" );
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document.write( "      g(x) =  x/ (over) x^2 + 9x +14   \r<BR>\n" );
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document.write( "Want us to find any values of x that are meaningless, specifically, that would make the denominator = 0. to do this make the denominator = 0\r<BR>\n" );
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document.write( "So, x^2 +9x+14 = 0\r<BR>\n" );
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document.write( "to show the way to solve it, they now show (x+2)(x+7)=0\r<BR>\n" );
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document.write( "Therefore, X= -2, -7\r<BR>\n" );
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document.write( "How do you get (x+2)(x+7) = 0 from x^2+9x+14+0?????? What happened here?  And where did the '9x' go ??? I am totally confused in how they worked out that -2 and -7 are meaningless. how did they get to this?<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 #134952 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=134952\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "\"Meaningless\" is really a misleading term in this context.  What you want to find are values of the variable that make the function undefined.  Any rational function is undefined for values of the variable that make a denominator equal zero.\r<BR>\n" );
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document.write( "So, we need to find values of x that satisfy <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x^2 + 9x + 14 = 0\">\r<BR>\n" );
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document.write( "If you have two binomial factors such as <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (x + 1)(x + 2)\"> you multiply them by using FOIL (First, Outside, Inside, Last), so:\r<BR>\n" );
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document.write( "First:  x * x = x^2\r<BR>\n" );
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document.write( "Outside x * 2 = 2x\r<BR>\n" );
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document.write( "Inside 1 * x = x\r<BR>\n" );
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document.write( "Last 1 * 2 = 2\r<BR>\n" );
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document.write( "Putting them all together:  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x^2 + 2x + x + 2 = x^2 + 3x + 2\">\r<BR>\n" );
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document.write( "In order to solve <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x^2 + 9x + 14 = 0\">, you need to work the FOIL process in reverse.\r<BR>\n" );
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document.write( "The trick is to find two numbers that multiplied together yield the constant term and when added together give the coefficient on the 1st degree term.\r<BR>\n" );
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document.write( "For the given equation, those numbers are 2 and 7.  2 times 7 is 14 and 2 plus 7 is 9.  So:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x^2 + 9x + 14 = (x + 2)(x + 7)\">.  Let's repeat the FOIL process to verify this is true.\r<BR>\n" );
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document.write( "First:  x^2  Outside:  +7x  Inside:  +2x   Last: +14, so <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x^2 + 9x + 14\">\r<BR>\n" );
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document.write( "Now that we are assured that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x^2 + 9x + 14 = (x + 2)(x + 7)\">, we can write:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (x + 2)(x + 7) = 0\">\r<BR>\n" );
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document.write( "Now we apply the Zero Product Rule.  The Zero Product Rule says that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large ab = 0\"> if and only if <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a = 0\"> or <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large b = 0\">.  Applying this to our problem:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (x + 2)(x + 7) = 0\"> if and only if <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (x + 2) = 0\"> or <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (x + 7) = 0\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x + 2 = 0\"> means <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x = -2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x + 7 = 0\"> means <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x = -7\">\r<BR>\n" );
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document.write( "Let's check this result by substituting these values into the equation we started with:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (-2)^2 + 9(-2) + 14 =  4 - 18 + 14 = 0\">, so -2 works.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (-7)^2 + 9(-7) + 14 = 49 - 63 + 14 = 0\">, so -7 works.\r<BR>\n" );
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document.write( "So, whenever <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x = -2\"> or <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x = -7\">, then the denominator of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large g(x) = \frac {x}{x^2 + 9x +14}\"> is equal to zero and the function is undefined. \r<BR>\n" );
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document.write( "Using set builder notation, you would describe the domain of <i>g</i> thus:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \{x | x \in \R, x \neq -2, x \neq -7\}\">\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
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