document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=242807>Question 242807</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The product of two consecutive integers is 71 more than their sum. Find the integers.\r<BR>\n" );
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document.write( "I think that the integers will be 71 plus the total of the two integers but I am really confused on how you will get that. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #177787 by </B><A HREF=/tutors/aboutme.mpl?userid=oberobic><B>oberobic(2304)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=oberobic><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=177787\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Setup what you know using the fewest unknowns.<BR>\n" );
document.write( "Two consecutive numbers could be called x & y.<BR>\n" );
document.write( "BUT, calling them x and x+1 is much easier and uses only one unknown.<BR>\n" );
document.write( "The product would be:<BR>\n" );
document.write( "x(x+1) = x^2+ x<BR>\n" );
document.write( "The sum would be:<BR>\n" );
document.write( "x + (x+1) = 2x + 1<BR>\n" );
document.write( "...<BR>\n" );
document.write( "We are told the product is 71 more than the sum, or the sum + 71 = product.<BR>\n" );
document.write( "Substituting the formulas we have developed<BR>\n" );
document.write( "2x + 1 + 71 = x^2 + x<BR>\n" );
document.write( "...<BR>\n" );
document.write( "Now we need to do the algebra to get it into standard form...<BR>\n" );
document.write( "Subtracting x^2 + x from both sides<BR>\n" );
document.write( "2x + 1 + 71 - (x^2 + x) = 0<BR>\n" );
document.write( "Simplify<BR>\n" );
document.write( "2x + 1 + 71 - x^2 - x = 0<BR>\n" );
document.write( "Rearranging and combining terms<BR>\n" );
document.write( "-x^2 + x + 72 = 0<BR>\n" );
document.write( "Multiplying both sides by -1 to make it look friendlier...<BR>\n" );
document.write( "x^2 - x - 72 = 0<BR>\n" );
document.write( "Factoring we arrive at<BR>\n" );
document.write( "(x + 8)(x - 9) = 0<BR>\n" );
document.write( "So x must be either -8 or 9.<BR>\n" );
document.write( "(Actually, algebra tells us that BOTH are answers.)<BR>\n" );
document.write( "...<BR>\n" );
document.write( "Checking our work is now the key to knowing the answer.<BR>\n" );
document.write( "If x = 9, then the two consecutive integers are 9 and 10.<BR>\n" );
document.write( "Does 2x + 1 + 71 = x^2 + x, when x = 9?<BR>\n" );
document.write( "2(9) + 1 + 71 (equals?) 9(9) + 9??<BR>\n" );
document.write( "18 + 1 + 71 = 18 + 72 = 9(2) + 9(8) = 9(10) = 90<BR>\n" );
document.write( "9(9) + 9 = 81 + 9 = 90<BR>\n" );
document.write( "So x = 9 works quite nicely.<BR>\n" );
document.write( "...<BR>\n" );
document.write( "What about x = -8?<BR>\n" );
document.write( "(ugly grunt work must be done.)<BR>\n" );
document.write( "Does 2x + 1 + 71 = x^2 + x, when x = -8?<BR>\n" );
document.write( "2(-8) + 1 + 71 = -16 + 1 + 71 = -16 + 72 = 56<BR>\n" );
document.write( "(-8)^2 + (-8) = 64 - 8 = 56<BR>\n" );
document.write( "56 = 56 is TRUE! (smile)<BR>\n" );
document.write( "...<BR>\n" );
document.write( "We have two answers.<BR>\n" );
document.write( "x = -8 or 9<BR>\n" );
document.write( "The two consecutive numbers are either 8 & 9 or -8 & -7.<BR>\n" );
document.write( "...<BR>\n" );
document.write( "Note that we are not too surprised by having two solutions because we have a QUADRATIC equation. </SPAN>\n" );
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