document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1169576>Question 1169576</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find all solutions to the system\r<BR>\n" );
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document.write( "a + b = 14<BR>\n" );
document.write( "a^3 + b^3 = 812.\r<BR>\n" );
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document.write( "How would I solve this? (Not looking for an answer by the way) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #794360 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=794360\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Recall that sum of cubes factoring formula is<BR>\n" );
document.write( "a^3 + b^3 = (a+b)(a^2 - ab + b^2)\r<BR>\n" );
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document.write( "Note how we have (a+b) show up. If we divide both sides by (a+b), then,\r<BR>\n" );
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document.write( "(a^3 + b^3)/(a+b) = [ (a+b)(a^2 - ab + b^2) ]/(a+b)<BR>\n" );
document.write( "(a^3 + b^3)/(a+b) = a^2 - ab + b^2<BR>\n" );
document.write( "a^2 - ab + b^2 = (a^3 + b^3)/(a+b)<BR>\n" );
document.write( "a^2 - ab + b^2 = (812)/(14)<BR>\n" );
document.write( "a^2 - ab + b^2 = 58<BR>\n" );
document.write( "We'll use this later. So let's call this equation 3.<BR>\n" );
document.write( "Note: since we divided by (a+b), we must require that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+%3C%3E+-b ALIGN=MIDDLE  ALT=\"a+%3C%3E+-b\"   DISABLED_style= \"max-width:90%; height:auto;\" > to avoid dividing by zero.\r<BR>\n" );
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document.write( "Go back to a+b = 14 and square both sides<BR>\n" );
document.write( "a+b = 14<BR>\n" );
document.write( "(a+b)^2 = 14^2<BR>\n" );
document.write( "a^2 + 2ab + b^2 = 196<BR>\n" );
document.write( "Call this equation 4\r<BR>\n" );
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document.write( "So we have equation 3 and equation 4 as<BR>\n" );
document.write( "a^2 - ab + b^2 = 58<BR>\n" );
document.write( "a^2 + 2ab + b^2 = 196<BR>\n" );
document.write( "If we subtract straight down, the a^2 and b^2 terms cancel out and go away. We end up with <BR>\n" );
document.write( "-3ab = -138<BR>\n" );
document.write( "ab = (-138)/(-3)<BR>\n" );
document.write( "ab = 46\r<BR>\n" );
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document.write( "So whatever 'a' and 'b' are, they must multiply to 46.<BR>\n" );
document.write( "They must also add to 14.\r<BR>\n" );
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document.write( "Consider the factorization<BR>\n" );
document.write( "(x-a)(x-b)\r<BR>\n" );
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document.write( "FOIL that out to get<BR>\n" );
document.write( "x^2 - ax - bx + ab<BR>\n" );
document.write( "x^2 - (a + b)x + ab\r<BR>\n" );
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document.write( "The roots of <BR>\n" );
document.write( "(x-a)(x-b) = 0<BR>\n" );
document.write( "are a and b\r<BR>\n" );
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document.write( "This means the roots of <BR>\n" );
document.write( "x^2 - (a + b)x + ab = 0<BR>\n" );
document.write( "are also a and b<BR>\n" );
document.write( "We have the roots adding to the negative of the middle coefficient; and also the roots multiplying to the last term. I'm using one of Vieta's Formulas.\r<BR>\n" );
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document.write( "Therefore, you need to solve<BR>\n" );
document.write( "x^2 - 14x + 46 = 0\r<BR>\n" );
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document.write( "I'll let you take over from here. Use the quadratic formula. <BR>\n" );
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