document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=790572>Question 790572</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A girl is 3 years younger than her brother. The product of their ages is 238. Find their ages.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #479317 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</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=479317\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The easiest way to the answer (in my opinion) is to factor 238.<BR>\n" );
document.write( "Dividing by 2, I find that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=238=2%2A119 ALIGN=MIDDLE  ALT=\"238=2%2A119\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "119 does not divide evenly by 2, 3, or 5, but it divides by 7:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=119=7%2A17 ALIGN=MIDDLE  ALT=\"119=7%2A17\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=238=2%2A7%2A17=14%2A17 ALIGN=MIDDLE  ALT=\"238=2%2A7%2A17=14%2A17\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The sister is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%2814%29 ALIGN=MIDDLE  ALT=\"highlight%2814%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the brother is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%2817%29 ALIGN=MIDDLE  ALT=\"highlight%2817%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "If you are studying prime factorization, or if you are not in advanced algebra, that is the way you are expected to solve the problem.<BR>\n" );
document.write( " <BR>\n" );
document.write( "If you are studying quadratic equations, you may be expected to write and solve one, in a long, painful, unnecessary waste of effort, like this:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >= age of the brother<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-3 ALIGN=MIDDLE  ALT=\"x-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >= age of the sister<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%28x-3%29=238 ALIGN=MIDDLE  ALT=\"x%28x-3%29=238\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-3x=238 ALIGN=MIDDLE  ALT=\"x%5E2-3x=238\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-3x-238=0 ALIGN=MIDDLE  ALT=\"x%5E2-3x-238=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That is the quadratic equation.<BR>\n" );
document.write( " <BR>\n" );
document.write( "There are 3 popular ways to solve such an equation.<BR>\n" );
document.write( "You could use factoring, \"completing the square\", or the quadratic formula.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Factoring involves finding two factors that multiply to yield 238, and differ by 3. In a way, that is what I did above, without even mentioning algebra or equations.<BR>\n" );
document.write( "Factoring, we find that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-3x-238=%28x-17%29%28x%2B14%29 ALIGN=MIDDLE  ALT=\"x%5E2-3x-238=%28x-17%29%28x%2B14%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > so we re-write the equation as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-17%29%28x%2B14%29=0 ALIGN=MIDDLE  ALT=\"%28x-17%29%28x%2B14%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "At that point we realize that a solution must make <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-17=0 ALIGN=MIDDLE  ALT=\"x-17=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B14=0 ALIGN=MIDDLE  ALT=\"x%2B14=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%2B14=0 ALIGN=MIDDLE  ALT=\"x%2B14=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-14 ALIGN=MIDDLE  ALT=\"x=-14\"   DISABLED_style= \"max-width:90%; height:auto;\" > which cannot be the brother's age, because it is a negative number.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-17=0 ALIGN=MIDDLE  ALT=\"x-17=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > leads to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=highlight%2817%29 ALIGN=MIDDLE  ALT=\"x=highlight%2817%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >= age of the brother, and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=17-3=highlight%2814%29 ALIGN=MIDDLE  ALT=\"17-3=highlight%2814%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >= age of the sister.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Completing the square would be:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-3x=238 ALIGN=MIDDLE  ALT=\"x%5E2-3x=238\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-3x%2B9%2F4=238%2B9%2F4 ALIGN=MIDDLE  ALT=\"x%5E2-3x%2B9%2F4=238%2B9%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-3%2F2%29%5E2=238%2B9%2F4 ALIGN=MIDDLE  ALT=\"%28x-3%2F2%29%5E2=238%2B9%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-3%2F2%29%5E2=952%2F4%2B9%2F4 ALIGN=MIDDLE  ALT=\"%28x-3%2F2%29%5E2=952%2F4%2B9%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-3%2F2%29%5E2=961%2F4 ALIGN=MIDDLE  ALT=\"%28x-3%2F2%29%5E2=961%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-3%2F2%29%5E2=%2831%2F2%29%5E2 ALIGN=MIDDLE  ALT=\"%28x-3%2F2%29%5E2=%2831%2F2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So either <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-3%2F2=-31%2F2 ALIGN=MIDDLE  ALT=\"x-3%2F2=-31%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-31%2F2%2B3%2F2%3C0 ALIGN=MIDDLE  ALT=\"x=-31%2F2%2B3%2F2%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (which cannot be the age of the brother), or<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-3%2F2=31%2F2 ALIGN=MIDDLE  ALT=\"x-3%2F2=31%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=31%2F2%2B3%2F2=34%2F2=highlight%2817%29 ALIGN=MIDDLE  ALT=\"x=31%2F2%2B3%2F2=34%2F2=highlight%2817%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >= age of the brother,<BR>\n" );
document.write( "so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=17-3=highlight%2814%29 ALIGN=MIDDLE  ALT=\"17-3=highlight%2814%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >= age of the sister.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Using the quadratic formula:<BR>\n" );
document.write( "The solutions to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be expressed as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "In the case of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-3x-238=0 ALIGN=MIDDLE  ALT=\"x%5E2-3x-238=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=1 ALIGN=MIDDLE  ALT=\"a=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-3 ALIGN=MIDDLE  ALT=\"b=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=-238 ALIGN=MIDDLE  ALT=\"c=-238\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-%28-3%29+%2B-+sqrt%28%28-3%29%5E2-4%2A1%2A%28-238%29+%29%29%2F%282%2A1%29=%283+%2B-+sqrt%289%2B952%29%29%2F2=%283+%2B-+sqrt%28961%29%29%2F2=3+%2B-+31%29%2F2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So the solutions to that equation are:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=%283-31%29%2F2=-28%2F2=-14 ALIGN=MIDDLE  ALT=\"c=%283-31%29%2F2=-28%2F2=-14\"   DISABLED_style= \"max-width:90%; height:auto;\" > (which is not a solution to the problem, because the brother's age cannot be negative),<BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%283%2B31%29%2F2=34%2F2=highlight%2817%29 ALIGN=MIDDLE  ALT=\"x=%283%2B31%29%2F2=34%2F2=highlight%2817%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >= age of the brother,<BR>\n" );
document.write( "so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=17-3=highlight%2814%29 ALIGN=MIDDLE  ALT=\"17-3=highlight%2814%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >= age of the sister.<BR>\n" );
document.write( " \r<BR>\n" );
document.write( "\n" );
document.write( "  </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );