document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209229>Question 1209229</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The real numbers a and b satisfy a - b = 2 and a^3 - b^3 = 8.\r<BR>\n" );
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document.write( "(a) Find all possible values of ab.<BR>\n" );
document.write( "(b) Find all possible values of a + b.<BR>\n" );
document.write( "(c) Find all possible values of a and b. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #847875 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=847875\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Part 1) Find all possible values of ab.\r<BR>\n" );
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document.write( "(a-b)^3 <BR>\n" );
document.write( "= (a-b)(a-b)^2<BR>\n" );
document.write( "= (a-b)(a^2-2ab+b^2)<BR>\n" );
document.write( "= a(a^2-2ab+b^2)-b(a^2-2ab+b^2)<BR>\n" );
document.write( "= (a^3-2a^2b+ab^2)+(-a^2b+2ab^2-b^3)<BR>\n" );
document.write( "= a^3 + (-2a^2b-a^2b) + (ab^2+2ab^2) - b^3<BR>\n" );
document.write( "= a^3 - 3a^2b + 3ab^2 - b^3<BR>\n" );
document.write( "= a^3-b^3-3ab(a-b)\r<BR>\n" );
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document.write( "In short,<BR>\n" );
document.write( "(a-b)^3 = a^3-b^3-3ab(a-b)<BR>\n" );
document.write( "You can skip over the previous paragraph if you have this formula memorized or written on a notecard. \r<BR>\n" );
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document.write( "Then we apply the equations <font color=green>a-b = 2</font> and <font color=blue>a^3-b^3 = 8</font> to isolate ab.<BR>\n" );
document.write( "So,<BR>\n" );
document.write( "(a-b)^3 = a^3-b^3-3ab(a-b)<BR>\n" );
document.write( "(<font color=green>a-b</font>)^3 = <font color=blue>a^3-b^3</font>-3ab(<font color=green>a-b</font>)<BR>\n" );
document.write( "(<font color=green>2</font>)^3 = <font color=blue>8</font>-3ab(<font color=green>2</font>) <BR>\n" );
document.write( "8 = 8 - 6ab<BR>\n" );
document.write( "-6ab = 8-8<BR>\n" );
document.write( "-6ab = 0<BR>\n" );
document.write( "ab = 0/(-6)<BR>\n" );
document.write( "<font color=red>ab = 0</font>\r<BR>\n" );
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document.write( "Part 2) Find all possible values of a+b.\r<BR>\n" );
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document.write( "c = a+b<BR>\n" );
document.write( "c^2 = a^2+2ab+b^2<BR>\n" );
document.write( "c^2 = a^2+2*0+b^2 ..... plug in ab = 0<BR>\n" );
document.write( "c^2 = a^2+b^2\r<BR>\n" );
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document.write( "Use the difference of cubes factoring formula<BR>\n" );
document.write( "a^3-b^3 = (a-b)(a^2+ab+b^2)<BR>\n" );
document.write( "a^3-b^3 = (a-b)(a^2+0+b^2)<BR>\n" );
document.write( "a^3-b^3 = (a-b)(a^2+b^2)<BR>\n" );
document.write( "<font color=blue>a^3-b^3</font> = (<font color=green>a-b</font>)c^2<BR>\n" );
document.write( "<font color=blue>8</font> = <font color=green>2</font>c^2<BR>\n" );
document.write( "c^2 = 8/2<BR>\n" );
document.write( "c^2 = 4<BR>\n" );
document.write( "c = sqrt(4) or c = -sqrt(4)<BR>\n" );
document.write( "c = 2 or c = -2<BR>\n" );
document.write( "<font color=red>a+b = 2 or a+b = -2</font>\r<BR>\n" );
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document.write( "Part 3) Find all possible values of a and b.\r<BR>\n" );
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document.write( "a-b = 2<BR>\n" );
document.write( "a = b+2\r<BR>\n" );
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document.write( "a^3 - b^3 = 8<BR>\n" );
document.write( "(b+2)^3 - b^3 = 8<BR>\n" );
document.write( "(b^3+3*b^2*2+3*b*2^2+2^3) - b^3 = 8<BR>\n" );
document.write( "6b^2+12b+8 = 8<BR>\n" );
document.write( "6b^2+12b = 0<BR>\n" );
document.write( "6b(b+2) = 0<BR>\n" );
document.write( "6b = 0 or b+2 = 0<BR>\n" );
document.write( "b = 0 or b = -2\r<BR>\n" );
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document.write( "If b = 0, then a = b+2 = 0+2 = 2<BR>\n" );
document.write( "One ordered pair solution is <font color=red>(a,b) = (2,0)</font>\r<BR>\n" );
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document.write( "If b = -2, then a = b+2 = -2+2 = 0<BR>\n" );
document.write( "The other ordered pair solution is <font color=red>(a,b) = (0,-2)</font>\r<BR>\n" );
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document.write( "Note that you can do part 3 first to determine a,b<BR>\n" );
document.write( "Then it's very easy to compute ab and a+b.\r<BR>\n" );
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document.write( "Answers:\r<BR>\n" );
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document.write( "<font color=red>ab = 0</font><BR>\n" );
document.write( "<font color=red>a+b = 2</font> or <font color=red>a+b = -2</font><BR>\n" );
document.write( "<font color=red>(a,b) = (2,0)</font> or  <font color=red>(a,b) = (0,-2)</font><BR>\n" );
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