document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199644>Question 1199644</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find three consecutive positive even integers such that the<BR>\n" );
document.write( "product of the median and largest integer is 6 less than 21 times<BR>\n" );
document.write( "the smallest integer.. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #833599 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3816)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=833599\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answer: <font size=4 color=red>14, 16, 18</font>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "=====================================================================\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Explanation:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Consecutive integers follow one after another. <BR>\n" );
document.write( "Example: 4, 5, 6<BR>\n" );
document.write( "Each adjacent neighboring item has a gap of 1.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Consecutive even integers are the same idea, but all of the values must be even. <BR>\n" );
document.write( "Example: 8, 10, 12<BR>\n" );
document.write( "Each adjacent neighboring item has a gap of 2.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let x be a positive even number from the set {2, 4, 6, 8, ...}\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x = 1st even integer<BR>\n" );
document.write( "x+2 = 2nd even integer = median = middle<BR>\n" );
document.write( "x+4 = 3rd even integer<BR>\n" );
document.write( "The gap from x to x+2 is +2, and the gap from x+2 to x+4 is also +2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(x+2)*(x+4) = \"product of median and the largest integer\"<BR>\n" );
document.write( "21x-6 = \"Six less than 21 times the smallest integer\" \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "\"The product of the median and largest integer is 6 less than 21 times the smallest integer\" translates to<BR>\n" );
document.write( "(x+2)*(x+4) = 21x-6\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We'll expand things out and get everything to one side<BR>\n" );
document.write( "(x+2)*(x+4) = 21x-6<BR>\n" );
document.write( "x^2+6x+8 = 21x-6<BR>\n" );
document.write( "x^2+6x+8-21x+6 = 0<BR>\n" );
document.write( "x^2-15x+14 = 0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Then you have a few options at this point.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "One method is to factor like so:<BR>\n" );
document.write( "x^2-15x+14 = 0<BR>\n" );
document.write( "(x-1)(x-14) = 0<BR>\n" );
document.write( "x-1 = 0 or x-14 = 0<BR>\n" );
document.write( "x = 1 or x = 14\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We ignore x = 1 since x must be even.<BR>\n" );
document.write( "Therefore, x = 14 is the only solution.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "-----------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Another method of solving:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We can apply the quadratic formula<BR>\n" );
document.write( "x^2-15x+14 = 0 is of the form ax^2+bx+c = 0<BR>\n" );
document.write( "where,<BR>\n" );
document.write( "a = 1<BR>\n" );
document.write( "b = -15<BR>\n" );
document.write( "c = 14<BR>\n" );
document.write( "So,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"x+=+%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\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+=+%28-%28-15%29%2B-sqrt%28%28-15%29%5E2-4%281%29%2814%29%29%29%2F%282%281%29%29 ALIGN=MIDDLE  ALT=\"x+=+%28-%28-15%29%2B-sqrt%28%28-15%29%5E2-4%281%29%2814%29%29%29%2F%282%281%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\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+=+%2815%2B-sqrt%28225-56%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"x+=+%2815%2B-sqrt%28225-56%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\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+=+%2815%2B-sqrt%28169%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"x+=+%2815%2B-sqrt%28169%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\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+=+%2815%2B-++13%29%2F%282%29 ALIGN=MIDDLE  ALT=\"x+=+%2815%2B-++13%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\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+=+%2815%2B13%29%2F%282%29 ALIGN=MIDDLE  ALT=\"x+=+%2815%2B13%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2815-13%29%2F%282%29 ALIGN=MIDDLE  ALT=\"x+=+%2815-13%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\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+=+%2828%29%2F%282%29 ALIGN=MIDDLE  ALT=\"x+=+%2828%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%282%29%2F%282%29 ALIGN=MIDDLE  ALT=\"x+=+%282%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\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+=+14 ALIGN=MIDDLE  ALT=\"x+=+14\"   DISABLED_style= \"max-width:90%; height:auto;\" > or  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+1 ALIGN=MIDDLE  ALT=\"x+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We arrive at the same two solutions found earlier. <BR>\n" );
document.write( "And as mentioned earlier, we ignore x = 1 to go for x = 14.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "-----------------------------------------------\r<BR>\n" );
document.write( "\n" );
document.write( "A third method:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Graph out y = x^2-15x+14 using a TI calculator, Desmos, or GeoGebra. <BR>\n" );
document.write( "There are tons of options out there so feel free to use your favorite graphing calculator.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28500%2C500%2C-3%2C16%2C-50%2C50%2C0%2Cx%5E2-15x%2B14%29 ALIGN=MIDDLE  ALT=\"graph%28500%2C500%2C-3%2C16%2C-50%2C50%2C0%2Cx%5E2-15x%2B14%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The parabola crosses the x axis at the locations (1,0) and (14,0) which shows that x = 1 and x = 14 are the two solutions or roots.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The term \"x intercept\" is another way of saying \"root\" or \"zero of a function\". \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "-----------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If x = 14, then,<BR>\n" );
document.write( "x+2 = 14+2 = 16<BR>\n" );
document.write( "x+4 = 14+4 = 18\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So that's how we arrive at <font size=4 color=red>14, 16, 18</font> as the final answer.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Check:<BR>\n" );
document.write( "median*largest = 16*18 = 288<BR>\n" );
document.write( "21*smallest-6 = 21*14-6 = 288<BR>\n" );
document.write( "This confirms that the equation (x+2)(x+4) = 21x-6 is correct when x = 14, and confirms the final answer.<BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );