document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=248047>Question 248047</A>: <I><SPAN STYLE=\"word-break: break-all;\"> the difference of two positive numbers is 3 and the sum of their square is 45. Find the numbers. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #180802 by </B><A HREF=/tutors/aboutme.mpl?userid=oberobic><B>oberobic(2304)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=oberobic><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=180802\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The difference of two positive numbers is 3.<BR>\n" );
document.write( "x - y = 3<BR>\n" );
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document.write( "x^2 + y^2 = 45<BR>\n" );
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document.write( "We can use the first equation to solve the second.<BR>\n" );
document.write( "x - y = 3<BR>\n" );
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document.write( "Add y to both sides<BR>\n" );
document.write( "x = y + 3<BR>\n" );
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document.write( "Substitute in the second equation and solve.<BR>\n" );
document.write( "x^2 + y^2 = 45<BR>\n" );
document.write( "(y+3)^2 + y^2 = 45<BR>\n" );
document.write( "(y^ + 6y + 9) + y^2 = 45<BR>\n" );
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document.write( "Collect terms<BR>\n" );
document.write( "2y^2 + 6y + 9 = 45<BR>\n" );
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document.write( "Subtract 45 from both sides.<BR>\n" );
document.write( "2y^2 + 6y - 36 = 0<BR>\n" );
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document.write( "Divide by 2<BR>\n" );
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document.write( "y^2 + 3y - 18 = 0<BR>\n" );
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document.write( "Can we factor 18 into two terms that are 3 apart?  Yes, 3 & 6.<BR>\n" );
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document.write( "(y + 6)(y - 3) = 0<BR>\n" );
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document.write( "So we have two candidate answers:  y = -6 and y = 3.<BR>\n" );
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document.write( "The problem says the numbers are positive, so we assume y = 3 is one answer.<BR>\n" );
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document.write( "Given x^2 + y^2 = 45, we know y^9 = 9.<BR>\n" );
document.write( "x^2 + 9 = 45<BR>\n" );
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document.write( "Subtract 9 from both sides<BR>\n" );
document.write( "x^2 = 36<BR>\n" );
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document.write( "Take the square root.<BR>\n" );
document.write( "x = 6.<BR>\n" );
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document.write( "So our second candidate answer is x = 6.<BR>\n" );
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document.write( "Checking our work.<BR>\n" );
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document.write( "What does x^2 + y^2 equal 45?<BR>\n" );
document.write( "3^2 + 6^2 = 9 + 36 = 45<BR>\n" );
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document.write( "Does x - y = 3?<BR>\n" );
document.write( "6-3 = 3<BR>\n" );
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document.write( "OK </SPAN>\n" );
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