document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=922274>Question 922274</A>: <I><SPAN STYLE=\"word-break: break-all;\">  	\r<BR>\n" );
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document.write( "An integer is four less than twice another.  If the product of the two integers is 70, one pair of integers that will satisfy the problem is _________.  (Give your answer as follows: 55 and 96) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #559464 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=559464\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 7 and 10; 10 is 4 less than twice 7.<BR>\n" );
document.write( "A quadratic equation may be expected, but it is not the most efficient way to solve the problem, and if you are going to solve the quadratic equation by factoring, why not just factor to get to the solution directly?<BR>\n" );
document.write( "After all we need a solution, so there is no need to make sure we find every possible solution.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Listing the pairs of factors is easy enough:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2A70=70 ALIGN=MIDDLE  ALT=\"1%2A70=70\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A35=70 ALIGN=MIDDLE  ALT=\"2%2A35=70\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2A14=70 ALIGN=MIDDLE  ALT=\"5%2A14=70\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2A10=70 ALIGN=MIDDLE  ALT=\"7%2A10=70\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The only pair where one factor is less than twice the other is 7 and 10.<BR>\n" );
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document.write( "NOTE:<BR>\n" );
document.write( "Finding the right pair of factors seems easy, and it is, because 70 is a nice number for this problem.<BR>\n" );
document.write( "The prime factoring of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=70 ALIGN=MIDDLE  ALT=\"70\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=70=2%2A5%2A7 ALIGN=MIDDLE  ALT=\"70=2%2A5%2A7\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That is a product of 3 prime factors, each with an exponent of 1.<BR>\n" );
document.write( "That means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=70 ALIGN=MIDDLE  ALT=\"70\"   DISABLED_style= \"max-width:90%; height:auto;\" > has <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2B1%29%2A%281%2B1%29%2A%281%2B1%29=2%2A3%2A3=8 ALIGN=MIDDLE  ALT=\"%281%2B1%29%2A%281%2B1%29%2A%281%2B1%29=2%2A3%2A3=8\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors in 4 pairs.<BR>\n" );
document.write( "Listing the pairs of factors is easy enough:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2A70=70 ALIGN=MIDDLE  ALT=\"1%2A70=70\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A35=70 ALIGN=MIDDLE  ALT=\"2%2A35=70\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2A14=70 ALIGN=MIDDLE  ALT=\"5%2A14=70\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2A10=70 ALIGN=MIDDLE  ALT=\"7%2A10=70\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The only pair where one factor is less than twice the other is 7 and 10.<BR>\n" );
document.write( " <BR>\n" );
document.write( "WITH THE QUADRATIC EQUATION, we can also list the fancier answer (-5 and -14):<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;\" >= one factor<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-4%29=2%28x-2%29 ALIGN=MIDDLE  ALT=\"2x-4%29=2%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >= the other factor (4 less than twice <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > )<BR>\n" );
document.write( "At this point, we can write our quadratic equation as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%282x-4%29=70 ALIGN=MIDDLE  ALT=\"x%282x-4%29=70\"   DISABLED_style= \"max-width:90%; height:auto;\" >--->{{2x^2-4x=70}}}---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2-4x-70=0 ALIGN=MIDDLE  ALT=\"2x%5E2-4x-70=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "or as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%28x-2%29x=70 ALIGN=MIDDLE  ALT=\"2%28x-2%29x=70\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%28x-2%29=35 ALIGN=MIDDLE  ALT=\"x%28x-2%29=35\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-2x=35 ALIGN=MIDDLE  ALT=\"x%5E2-2x=35\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-2x-35=0 ALIGN=MIDDLE  ALT=\"x%5E2-2x-35=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "You can apply the quadratic formula to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2-4x-70=0 ALIGN=MIDDLE  ALT=\"2x%5E2-4x-70=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-2x-35=0 ALIGN=MIDDLE  ALT=\"x%5E2-2x-35=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "You can solve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-2x-35=0 ALIGN=MIDDLE  ALT=\"x%5E2-2x-35=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > by factoring or by completing the square.<BR>\n" );
document.write( "You can also realize that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%28x-2%29=35 ALIGN=MIDDLE  ALT=\"x%28x-2%29=35\"   DISABLED_style= \"max-width:90%; height:auto;\" > screams <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=7 ALIGN=MIDDLE  ALT=\"x=7\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "Completing the square:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-2x=35 ALIGN=MIDDLE  ALT=\"x%5E2-2x=35\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-2x%2B1=35%2B1 ALIGN=MIDDLE  ALT=\"x%5E2-2x%2B1=35%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%29%5E2=36 ALIGN=MIDDLE  ALT=\"%28x-1%29%5E2=36\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x-1=6%2C%22or%22%2Cx-1=-6%29 ALIGN=MIDDLE  ALT=\"system%28x-1=6%2C%22or%22%2Cx-1=-6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=6%2B1=7%2C%22or%22%2Cx=-6%2B1=-5%29 ALIGN=MIDDLE  ALT=\"system%28x=6%2B1=7%2C%22or%22%2Cx=-6%2B1=-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "With <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=7 ALIGN=MIDDLE  ALT=\"x=7\"   DISABLED_style= \"max-width:90%; height:auto;\" > , the other number is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-4=2%2A7-4=10 ALIGN=MIDDLE  ALT=\"2x-4=2%2A7-4=10\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "If we want to be fancy, we say that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-5 ALIGN=MIDDLE  ALT=\"x=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > is an integer, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-4=2%2A%28-5%29-4=-10-4=-14 ALIGN=MIDDLE  ALT=\"2x-4=2%2A%28-5%29-4=-10-4=-14\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the other integer.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Solving by factoring:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-2x-35=0 ALIGN=MIDDLE  ALT=\"x%5E2-2x-35=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-7%29%28x%2B5%29=0 ALIGN=MIDDLE  ALT=\"%28x-7%29%28x%2B5%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x-7=0%2C%22or%22%2Cx%2B5=0%29 ALIGN=MIDDLE  ALT=\"system%28x-7=0%2C%22or%22%2Cx%2B5=0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=7%2C%22or%22%2Cx=-5%29 ALIGN=MIDDLE  ALT=\"system%28x=7%2C%22or%22%2Cx=-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "Applying the quadratic formula to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2-4x-70=0 ALIGN=MIDDLE  ALT=\"2x%5E2-4x-70=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=2 ALIGN=MIDDLE  ALT=\"a=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-4 ALIGN=MIDDLE  ALT=\"b=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=-70 ALIGN=MIDDLE  ALT=\"c=-70\"   DISABLED_style= \"max-width:90%; height:auto;\" ><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;\" > turns into<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-%28-4%29+%2B-+sqrt%28%28-4%29%5E2-4%2A2%2A%28-70%29%29%29%2F%282%2A2%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-%28-4%29+%2B-+sqrt%28%28-4%29%5E2-4%2A2%2A%28-70%29%29%29%2F%282%2A2%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%284+%2B-+sqrt%2816%2B560%29%29%2F4+ ALIGN=MIDDLE  ALT=\"x+=+%284+%2B-+sqrt%2816%2B560%29%29%2F4+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%284+%2B-+sqrt%28576%29%29%2F4+ ALIGN=MIDDLE  ALT=\"x+=+%284+%2B-+sqrt%28576%29%29%2F4+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%284+%2B-+24%29%2F4+ ALIGN=MIDDLE  ALT=\"x+=+%284+%2B-+24%29%2F4+\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=%284%2B24%29%2F4=28%2F4=7%2C%22or%22%2Cx=%284-24%29%2F4=%28-20%29%2F4=-5%29 ALIGN=MIDDLE  ALT=\"system%28x=%284%2B24%29%2F4=28%2F4=7%2C%22or%22%2Cx=%284-24%29%2F4=%28-20%29%2F4=-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );