document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209394>Question 1209394</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let x and y be complex numbers.  If x + y = 2 and xy = 7 - x^2 - y^2, then what is x^2 + y^2? </SPAN>\n" );
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document.write( "Answer: <font color=red>10</font>\r<BR>\n" );
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document.write( "Work Shown<BR>\n" );
document.write( "xy = 7 - x^2 - y^2<BR>\n" );
document.write( "xy = 7 - (x^2 + y^2)<BR>\n" );
document.write( "x^2 + y^2 = 7 - xy<BR>\n" );
document.write( "2(x^2 + y^2) = 14 - 2xy<BR>\n" );
document.write( "Let's call this equation (3)\r<BR>\n" );
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document.write( "x + y = 2<BR>\n" );
document.write( "(x + y)^2 = 2^2<BR>\n" );
document.write( "x^2 + 2xy + y^2 = 4<BR>\n" );
document.write( "x^2 + y^2 = 4 - 2xy<BR>\n" );
document.write( "Let's call this equation (4)\r<BR>\n" );
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document.write( "Subtract straight down for equations (3) and (4) to arrive at<BR>\n" );
document.write( "x^2 + y^2 = <font color=red>10</font>\r<BR>\n" );
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document.write( "An alternative approach would be to rearrange x+y = 2 into y = 2-x<BR>\n" );
document.write( "Then,<BR>\n" );
document.write( "xy = 7-x^2-y^2<BR>\n" );
document.write( "x(2-x) = 7-x^2-(2-x)^2<BR>\n" );
document.write( "Solving that for x yields x = -1 and x = 3<BR>\n" );
document.write( "I'll let the student handle the scratch work.\r<BR>\n" );
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document.write( "If x = -1 then y = 2-x = 2-(-1) = 3<BR>\n" );
document.write( "If x = 3 then y = 2-x = 2-3 = -1<BR>\n" );
document.write( "We get the same pair of values just in a different order.<BR>\n" );
document.write( "Turns out the order doesn't matter.\r<BR>\n" );
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document.write( "Then,<BR>\n" );
document.write( "x^2 + y^2 = (-1)^2 + (3)^2 = <font color=red>10</font><BR>\n" );
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