document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208880>Question 1208880</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If (a + 1)(b + 1)(a + b) = 1530<BR>\n" );
document.write( "and a³ + b³ = 1241<BR>\n" );
document.write( "then a + b = ? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #847388 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</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=847388\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "The other tutor has a great solution. <BR>\n" );
document.write( "Perhaps it's the most efficient pathway to solve. <BR>\n" );
document.write( "I'll provide another method.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "w = a+b<BR>\n" );
document.write( "(a+1)(b+1)(a+b) = 1530<BR>\n" );
document.write( "(ab+a+b+1)(a+b) = 1530<BR>\n" );
document.write( "(ab+w+1)w = 1530<BR>\n" );
document.write( "ab+w+1 = 1530/w<BR>\n" );
document.write( "ab = 1530/w - w - 1<BR>\n" );
document.write( "Let's call this equation (1) to use later. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Then notice how,<BR>\n" );
document.write( "w = a+b<BR>\n" );
document.write( "w^2 = (a+b)^2<BR>\n" );
document.write( "w^2 = a^2+2ab+b^2<BR>\n" );
document.write( "w^2-3ab = a^2+2ab+b^2-3ab<BR>\n" );
document.write( "w^2-3ab = a^2-ab+b^2<BR>\n" );
document.write( "Let's call this equation (2)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Use the sum of cubes formula to say the following<BR>\n" );
document.write( "a^3+b^3 = (a+b)(a^2-ab+b^2)<BR>\n" );
document.write( "a^3+b^3 = w(w^2-3ab) ............................. plug in equation (2)<BR>\n" );
document.write( "1241 = w(w^2-3*( 1530/w - w - 1 )) ............... plug in equation (1)<BR>\n" );
document.write( "1241 = w^3-3w*( 1530/w) + 3w^2 + 3w<BR>\n" );
document.write( "1241 = w^3-4590 + 3w^2 + 3w<BR>\n" );
document.write( "w^3-4590 + 3w^2 + 3w-1241 = 0<BR>\n" );
document.write( "w^3 + 3w^2 + 3w - 5831 = 0<BR>\n" );
document.write( "From here you can use a graphing calculator to graph y = x^3 + 3x^2 + 3x - 5831<BR>\n" );
document.write( "The only real number root is x = 17. This is where the cubic curve crosses the x axis. You'll likely need to adjust your window to be able to see this root.<BR>\n" );
document.write( "As a check,<BR>\n" );
document.write( "x^3 + 3x^2 + 3x - 5831 = 0<BR>\n" );
document.write( "(17)^3 + 3(17)^2 + 3(17) - 5831 = 0<BR>\n" );
document.write( "4913 + 867 + 51 - 5831 = 0<BR>\n" );
document.write( "0 = 0<BR>\n" );
document.write( "This proves that w = 17 is a solution to w^3 + 3w^2 + 3w - 5831 = 0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Through trial and error (aided with a spreadsheet), you can determine that a = 8 and b = 9 satisfy the two given equations.<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >(a+1)*(b+1)*(a+b) = 1530</br>(8+1)*(9+1)*(8+9) = 1530</br>9*10*17 = 1530</br>1530 = 1530 .... works</br></TD><TD >a^3+b^3 = 1241</br>8^3+9^3 = 1241</br>512+729 = 1241</br>1241 = 1241 .... works</br></TD></tr>\n" );
document.write( "</TABLE>So we conclude that a+b = 8+9 = 17<BR>\n" );
document.write( "Due to symmetry, we could also say that a = 9 and b = 8.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Answer: <font color=red>17</font><BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );