document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=81771>Question 81771</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Problem<BR>\n" );
document.write( "Find three consecutive integers such that the sum of their squares is 77.<BR>\n" );
document.write( "This is what I have, but not sure.<BR>\n" );
document.write( "x^2 + x^2 + x^2 = 77 I don't know where to go from here. Thank you for your help. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #58599 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=58599\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The first thing to do is to let x be the first integer. Then the next consecutive integer<BR>\n" );
document.write( "is 1 number higher or x + 1. And the next consecutive integer after that is (x+1) + 1 or<BR>\n" );
document.write( "x + 2.<BR>\n" );
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document.write( "The squares of these numbers are:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%29 ALIGN=MIDDLE  ALT=\"%28x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x+%2B+1%29%5E2+=+x%5E2+%2B+2x+%2B+1 ALIGN=MIDDLE  ALT=\"%28x+%2B+1%29%5E2+=+x%5E2+%2B+2x+%2B+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x+%2B+2%29%5E2+=+x%5E2+%2B+4x+%2B+4 ALIGN=MIDDLE  ALT=\"%28x+%2B+2%29%5E2+=+x%5E2+%2B+4x+%2B+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "If you add these three squares together you 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+2x+%2B+1+%2B+x%5E2+%2B+4x+%2B+4 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+x%5E2+%2B+2x+%2B+1+%2B+x%5E2+%2B+4x+%2B+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Combine the like terms and you get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2+%2B%282+%2B+4%29x+%2B+%281+%2B+4%29 ALIGN=MIDDLE  ALT=\"3x%5E2+%2B%282+%2B+4%29x+%2B+%281+%2B+4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This reduces to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2+%2B+6x+%2B+5 ALIGN=MIDDLE  ALT=\"3x%5E2+%2B+6x+%2B+5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "This is the sum of the squares and the problem tells you that this sum equals 77. So<BR>\n" );
document.write( "the equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2+%2B+6x+%2B+5+=+77 ALIGN=MIDDLE  ALT=\"3x%5E2+%2B+6x+%2B+5+=+77\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Put this into standard quadratic form by subtracting 77 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=3x%5E2+%2B+6x+-+72+=+0 ALIGN=MIDDLE  ALT=\"3x%5E2+%2B+6x+-+72+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Since this is in standard quadratic form, you can solve it by using the quadratic<BR>\n" );
document.write( "formula. Although it may be difficult to recognize, the left side of this equation can<BR>\n" );
document.write( "be factored:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%28x+-+4%29%28x+%2B+6%29+=+0 ALIGN=MIDDLE  ALT=\"3%28x+-+4%29%28x+%2B+6%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Divide both sides by 3 to reduce the equation to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x+-+4%29%28x+%2B+6%29+=+0 ALIGN=MIDDLE  ALT=\"%28x+-+4%29%28x+%2B+6%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This equation will be true if one of the factors is zero because a multiplier of zero on the<BR>\n" );
document.write( "left side will make the left side equal to zero, just as the right side is.<BR>\n" );
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document.write( "So solve this equation by 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+-+4+=+0 ALIGN=MIDDLE  ALT=\"x+-+4+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Solve by adding 4 to both sides to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+4 ALIGN=MIDDLE  ALT=\"x+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This would mean that the next two <BR>\n" );
document.write( "consecutive integers are 5 and 6.<BR>\n" );
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document.write( "Now set the second factor equal to zero:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x+%2B+6%29+=+0 ALIGN=MIDDLE  ALT=\"%28x+%2B+6%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Solve by subtracting 6 from both sides to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-+6 ALIGN=MIDDLE  ALT=\"x+=+-+6\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  This makes the next two<BR>\n" );
document.write( "consecutive integers (x+1 and x + 2) equal to -5 and -4.<BR>\n" );
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document.write( "So we have two possibilities. One solution is 4, 5, and 6. The other solution is -6, -5,<BR>\n" );
document.write( "and -4.<BR>\n" );
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document.write( "Let's check them out. <BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%5E2+%2B+5%5E2+%2B+6%5E2+=+16+%2B+25+%2B+36+=+77 ALIGN=MIDDLE  ALT=\"4%5E2+%2B+5%5E2+%2B+6%5E2+=+16+%2B+25+%2B+36+=+77\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "That works. Now:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-6%29%5E2+%2B+%28-5%29%5E2+%2B+%28-4%29%5E2+=+36+%2B+25+%2B+16+=+77 ALIGN=MIDDLE  ALT=\"%28-6%29%5E2+%2B+%28-5%29%5E2+%2B+%28-4%29%5E2+=+36+%2B+25+%2B+16+=+77\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Although it's a little more difficult to think about, -6, -5, and -4 are three consecutive<BR>\n" );
document.write( "integers that also work.<BR>\n" );
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document.write( "Hope this helps you to understand the problem.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );