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 #58595 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=58595\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let x=1st #, y=2nd #, z=3rd #\r<BR>\n" );
document.write( "\n" );
document.write( "Since you have consecutive integers, this means the integers are right next to each other. For instance, 1,2, and 3 are all consecutive integers. If x=1, then y=2, and z=3. So if you wanted them all in terms of x, you would say:\r<BR>\n" );
document.write( "\n" );
document.write( "x=1, y=x+1, z=x+2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since if you plug in x=1 into x+2, you get z=3. So the same applies to this problem. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So now we're adding the squares of these numbers to get 77 like this:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2By%5E2%2Bz%5E2=77 ALIGN=MIDDLE  ALT=\"x%5E2%2By%5E2%2Bz%5E2=77\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "But we can write y as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B1 ALIGN=MIDDLE  ALT=\"x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and z as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B2 ALIGN=MIDDLE  ALT=\"x%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" > so we can replace y and z:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B%28x%2B1%29%5E2%2B%28x%2B2%29%5E2=77 ALIGN=MIDDLE  ALT=\"x%5E2%2B%28x%2B1%29%5E2%2B%28x%2B2%29%5E2=77\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2Bx%5E2%2B2x%2B1%2Bx%5E2%2B4x%2B4=77 ALIGN=MIDDLE  ALT=\"x%5E2%2Bx%5E2%2B2x%2B1%2Bx%5E2%2B4x%2B4=77\"   DISABLED_style= \"max-width:90%; height:auto;\" > Foil the two parenthesis separately\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2%2B6x%2B5=77 ALIGN=MIDDLE  ALT=\"3x%5E2%2B6x%2B5=77\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2%2B6x%2B5-77=cross%2877-77%29 ALIGN=MIDDLE  ALT=\"3x%5E2%2B6x%2B5-77=cross%2877-77%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract 77 from both sides\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2%2B6x-72=0 ALIGN=MIDDLE  ALT=\"3x%5E2%2B6x-72=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now that we have a quadratic, lets use the quadratic formula to solve for x:\r<BR>\n" );
document.write( "\n" );
document.write( "<TABLE  BORDER=1>\n" );
document.write( "\n" );
document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=quadratic.solver>SOLVE quadratic equation with variable</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Quadratic equation <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;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2%2B6x%2B-72+=+0 ALIGN=MIDDLE  ALT=\"3x%5E2%2B6x%2B-72+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) has the following solutons:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  For these solutions to exist, the <U>discriminant</U> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > should not be a negative number.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  First, we need to compute the discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac=%286%29%5E2-4%2A3%2A-72=900 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%286%29%5E2-4%2A3%2A-72=900\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=900 is greater than zero. That means that there are two solutions: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5B12%5D+=+%28-6%2B-sqrt%28+900+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28-6%2B-sqrt%28+900+%29%29%2F2%5Ca\"   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=x%5B1%5D+=+%28-%286%29%2Bsqrt%28+900+%29%29%2F2%5C3+=+4 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%286%29%2Bsqrt%28+900+%29%29%2F2%5C3+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D+=+%28-%286%29-sqrt%28+900+%29%29%2F2%5C3+=+-6 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%286%29-sqrt%28+900+%29%29%2F2%5C3+=+-6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <BR>\n" );
document.write( "    Quadratic expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2%2B6x%2B-72 ALIGN=MIDDLE  ALT=\"3x%5E2%2B6x%2B-72\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be factored:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2%2B6x%2B-72+=+3%28x-4%29%2A%28x--6%29 ALIGN=MIDDLE  ALT=\"3x%5E2%2B6x%2B-72+=+3%28x-4%29%2A%28x--6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 4, -6.\n" );
document.write( "Here's your graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+3%2Ax%5E2%2B6%2Ax%2B-72+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+3%2Ax%5E2%2B6%2Ax%2B-72+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So we have 2 possible numbers for our first number. <BR>\n" );
document.write( "Now lets check our first possible answer x=4:\r<BR>\n" );
document.write( "\n" );
document.write( "Lets find our 2nd number using x=4:\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%2B1 ALIGN=MIDDLE  ALT=\"y=x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=4%2B1 ALIGN=MIDDLE  ALT=\"y=4%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in x=4\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=5 ALIGN=MIDDLE  ALT=\"y=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "So our 2nd number is 5\r<BR>\n" );
document.write( "\n" );
document.write( "Lets find our 3rd number using x=4\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=x%2B2 ALIGN=MIDDLE  ALT=\"z=x%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=4%2B2 ALIGN=MIDDLE  ALT=\"z=4%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in x=4\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=6 ALIGN=MIDDLE  ALT=\"z=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "So our 3rd number is 6\r<BR>\n" );
document.write( "\n" );
document.write( "So our 3 numbers are 4,5,6\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Check:\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%5E2%2B5%5E2%2B6%5E2=77 ALIGN=MIDDLE  ALT=\"4%5E2%2B5%5E2%2B6%5E2=77\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in x=4,y=5,z=6\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16%2B25%2B36=77 ALIGN=MIDDLE  ALT=\"16%2B25%2B36=77\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=77=77 ALIGN=MIDDLE  ALT=\"77=77\"   DISABLED_style= \"max-width:90%; height:auto;\" > works\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now lets check our other possible answer x=-6:\r<BR>\n" );
document.write( "\n" );
document.write( "Lets find the 2nd number using x=-6\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%2B1 ALIGN=MIDDLE  ALT=\"y=x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-6%2B1 ALIGN=MIDDLE  ALT=\"y=-6%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in x=-6\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-5 ALIGN=MIDDLE  ALT=\"y=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "So our 2nd number is -5\r<BR>\n" );
document.write( "\n" );
document.write( "Lets find our 3rd number using x=-6\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=x%2B2 ALIGN=MIDDLE  ALT=\"z=x%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=-6%2B2 ALIGN=MIDDLE  ALT=\"z=-6%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in x=-6\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=-4 ALIGN=MIDDLE  ALT=\"z=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "So our 3rd number is -4\r<BR>\n" );
document.write( "\n" );
document.write( "So our 3 numbers are -6,-5,-4\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Check:\r<BR>\n" );
document.write( "\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=77 ALIGN=MIDDLE  ALT=\"%28-6%29%5E2%2B%28-5%29%5E2%2B%28-4%29%5E2=77\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in x=-6,y=-5,z=-4\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=36%2B25%2B16=77 ALIGN=MIDDLE  ALT=\"36%2B25%2B16=77\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=77=77 ALIGN=MIDDLE  ALT=\"77=77\"   DISABLED_style= \"max-width:90%; height:auto;\" > works\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So we have 2 sets of numbers that work. We have one set\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "4,5,6  \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "and we also have\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "-6,-5,-4\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );