document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=110012>Question 110012</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0395977223/algebrahomeworkh>algebra(structure and method)</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> the sum of the squares of two consecutive positive even integers is 340. find the integers. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #80194 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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=80194\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Begin by letting x equal one of the positive even integers. Then the next consecutive<BR>\n" );
document.write( "positive even integer is x + 2 (because even numbers are 2 units away from the preceding<BR>\n" );
document.write( "even integer.)<BR>\n" );
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document.write( "Square the two integers to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B2%29%5E2+=+x%5E2+%2B+4x+%2B+4 ALIGN=MIDDLE  ALT=\"%28x%2B2%29%5E2+=+x%5E2+%2B+4x+%2B+4\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Then add these<BR>\n" );
document.write( "two squares and get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+x%5E2+%2B+4x+%2B+4 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+x%5E2+%2B+4x+%2B+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "You can combine the two <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > terms to simplify this to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2+%2B+4x+%2B+4 ALIGN=MIDDLE  ALT=\"2x%5E2+%2B+4x+%2B+4\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
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document.write( "and you are told that this sum equals 340. In equation form this is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2+%2B+4x+%2B+4+=+340 ALIGN=MIDDLE  ALT=\"2x%5E2+%2B+4x+%2B+4+=+340\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "You can simplify this a little by dividing both sides (each term) by 2 because 2 is a factor<BR>\n" );
document.write( "of all the terms in this equation. Dividing by 2 results in:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+2x+%2B+2+=+170 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+2x+%2B+2+=+170\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Get this into standard quadratic form by subtracting 170 from both sides of the equation so<BR>\n" );
document.write( "that the right side is zero. This subtraction leads to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+2x+-+168+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+2x+-+168+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "The left side of this equation can be factored. This factoring leads to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x+%2B+14%29%2A%28x+-+12%29+=+0 ALIGN=MIDDLE  ALT=\"%28x+%2B+14%29%2A%28x+-+12%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This equation will be true if either of the factors is zero, because a multiplication<BR>\n" );
document.write( "by zero on the left side results in the entire left side becoming zero and therefore<BR>\n" );
document.write( "equal to the right side.  So find out what values of x cause a factor to equal zero by<BR>\n" );
document.write( "setting each factor equal to zero:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x+%2B+14+=+0 ALIGN=MIDDLE  ALT=\"+x+%2B+14+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Solve by subtracting 14 from both sides to get:<BR>\n" );
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document.write( "<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;\" ><BR>\n" );
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document.write( "So one of the integers that could satisfy this equation is -14, but we can eliminate<BR>\n" );
document.write( "this because the problem requires that the consecutive even integers be positive. <BR>\n" );
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document.write( "Setting the other factor equal to zero:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+-+12+=+0 ALIGN=MIDDLE  ALT=\"x+-+12+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Solve for x by adding 12 to both sides to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+12 ALIGN=MIDDLE  ALT=\"x+=+12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This answer tells us that the first positive even integer is 12, and therefore, then next<BR>\n" );
document.write( "consecutive even integer is 14.<BR>\n" );
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document.write( "Check these answers. Does <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%5E2+%2B+14%5E2+=+340 ALIGN=MIDDLE  ALT=\"12%5E2+%2B+14%5E2+=+340\"   DISABLED_style= \"max-width:90%; height:auto;\" >? Yes it does because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%5E2+=+144 ALIGN=MIDDLE  ALT=\"12%5E2+=+144\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=14%5E2+=+196 ALIGN=MIDDLE  ALT=\"14%5E2+=+196\"   DISABLED_style= \"max-width:90%; height:auto;\" >. <BR>\n" );
document.write( "So our check is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=144+%2B+196+=+340 ALIGN=MIDDLE  ALT=\"144+%2B+196+=+340\"   DISABLED_style= \"max-width:90%; height:auto;\" > and this means that our answer works. <BR>\n" );
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document.write( "The answer to this problem is that two consecutive positive even integers that work are 12 and 14<BR>\n" );
document.write( "because when their squares are summed, the total is 340.<BR>\n" );
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document.write( "Hope this helps you to understand the problem.<BR>\n" );
document.write( ". </SPAN>\n" );
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