document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209398>Question 1209398</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let a and b be complex numbers.  If a + b = 4 and a^2 + b^2 = 6 + ab, then what is a^3 + b^3? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848751 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=848751\">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 color=red>24</font>\r<BR>\n" );
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document.write( "Explanation<BR>\n" );
document.write( "I'll show 3 methods to solving this problem.<BR>\n" );
document.write( "There could be other pathways.\r<BR>\n" );
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document.write( "Method 1\r<BR>\n" );
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document.write( "The given equations are<BR>\n" );
document.write( "a+b = 4<BR>\n" );
document.write( "a^2+b^2 = 6+ab<BR>\n" );
document.write( "Let's refer to these as equations (1) and (2) in the order presented.\r<BR>\n" );
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document.write( "a^3+b^3 = (a+b)*(a^2-ab+b^2) ...... sum of cubes factoring formula<BR>\n" );
document.write( "a^3+b^3 = (a+b)*(a^2+b^2-ab)<BR>\n" );
document.write( "a^3+b^3 = (a+b)*(6+ab-ab) ....... substitute in equation (2)<BR>\n" );
document.write( "a^3+b^3 = (a+b)*(6)<BR>\n" );
document.write( "a^3+b^3 = (4)*(6) ...... substitute in equation (1)<BR>\n" );
document.write( "a^3+b^3 = <font color=red>24</font>\r<BR>\n" );
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document.write( "Method 2\r<BR>\n" );
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document.write( "Square both sides of equation (1)<BR>\n" );
document.write( "a+b = 4<BR>\n" );
document.write( "(a+b)^2 = 4^2<BR>\n" );
document.write( "a^2+2ab+b^2 = 16<BR>\n" );
document.write( "(a^2+b^2)+2ab = 16<BR>\n" );
document.write( "(6+ab)+2ab = 16 ..... substitute in equation (2)<BR>\n" );
document.write( "6+3ab = 16<BR>\n" );
document.write( "3ab = 16-6<BR>\n" );
document.write( "3ab = 10<BR>\n" );
document.write( "Let's call this equation (3)\r<BR>\n" );
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document.write( "Now cube both sides of equation (1)<BR>\n" );
document.write( "a+b = 4<BR>\n" );
document.write( "(a+b)^3 = 4^3<BR>\n" );
document.write( "a^3 + 3a^2b + 3ab^2 + b^3 = 64 ... use binomial expansion formula<BR>\n" );
document.write( "a^3 + b^3 + 3a^2b + 3ab^2 = 64<BR>\n" );
document.write( "a^3 + b^3 + 3ab(a+b) = 64<BR>\n" );
document.write( "a^3 + b^3 + 10*(4) = 64 .... substitute in equations (1) and (3)<BR>\n" );
document.write( "a^3 + b^3 + 40 = 64<BR>\n" );
document.write( "a^3 + b^3 = 64-40<BR>\n" );
document.write( "a^3 + b^3 = <font color=red>24</font> is the final answer.\r<BR>\n" );
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document.write( "Method 3\r<BR>\n" );
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document.write( "a+b = 4 rearranges into a = 4-b<BR>\n" );
document.write( "Substitute that into a^2 + b^2 = 6 + ab and you'll get (4-b)^2 + b^2 = 6 + (4-b)*b<BR>\n" );
document.write( "Expand everything out and get everything to one side.<BR>\n" );
document.write( "You should arrive at 3b^2-12b+10 = 0<BR>\n" );
document.write( "I'll skip steps and only provide the key milestones. \r<BR>\n" );
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document.write( "Use of the quadratic formula yields the roots <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+%286%2Bsqrt%286%29%29%2F3 ALIGN=MIDDLE  ALT=\"b+=+%286%2Bsqrt%286%29%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+%286-sqrt%286%29%29%2F3 ALIGN=MIDDLE  ALT=\"b+=+%286-sqrt%286%29%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "If b is one of those roots then a = 4-b is the other root. <BR>\n" );
document.write( "The order doesn't matter.\r<BR>\n" );
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document.write( "So you would get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+%286%2Bsqrt%286%29%29%2F3 ALIGN=MIDDLE  ALT=\"a+=+%286%2Bsqrt%286%29%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+%286-sqrt%286%29%29%2F3 ALIGN=MIDDLE  ALT=\"b+=+%286-sqrt%286%29%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > in either order.\r<BR>\n" );
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document.write( "Cubing both of those produces <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E3+=+%282%2F9%29%2A%2854%2B19%2Asqrt%286%29%29 ALIGN=MIDDLE  ALT=\"a%5E3+=+%282%2F9%29%2A%2854%2B19%2Asqrt%286%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E3+=+%282%2F9%29%2A%2854-19%2Asqrt%286%29%29 ALIGN=MIDDLE  ALT=\"b%5E3+=+%282%2F9%29%2A%2854-19%2Asqrt%286%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The sum of which is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E3%2Bb%5E3+=+%284%2F9%29%2A54+=+24 ALIGN=MIDDLE  ALT=\"a%5E3%2Bb%5E3+=+%284%2F9%29%2A54+=+24\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Note the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%286%29 ALIGN=MIDDLE  ALT=\"sqrt%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > terms cancel out.\r<BR>\n" );
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document.write( "A somewhat similar question is found <a  rel=nofollow HREF=\"https://www.algebra.com/algebra/homework/coordinate/Linear-systems.faq.question.1209394.html\">here</a><BR>\n" );
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