document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1201936>Question 1201936</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If p + 1/p = 5 and p does not equal 0, which of the following is a possible value of p - 1/p ?\r<BR>\n" );
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document.write( "(A) sqrt(25)<BR>\n" );
document.write( "(B) sqrt(24)<BR>\n" );
document.write( "(C) sqrt(23)<BR>\n" );
document.write( "(D) sqrt(22)<BR>\n" );
document.write( "(E) sqrt(21) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #836517 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=836517\">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 size=4>Choice (E)  sqrt(21)</font>\r<BR>\n" );
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document.write( "Explanation:\r<BR>\n" );
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document.write( "Square both sides of the given equation<BR>\n" );
document.write( "p+ (1/p) = 5<BR>\n" );
document.write( "[ p+ (1/p) ]^2 = 5^2<BR>\n" );
document.write( "[ p+ (1/p) ][ p+ (1/p) ] = 25<BR>\n" );
document.write( "p*p + p*(1/p) + (1/p)*p + (1/p)(1/p) = 25 .... FOIL rule<BR>\n" );
document.write( "p^2 + 1 + 1 + (1/p)^2 = 25<BR>\n" );
document.write( "p^2 + 2 + (1/p)^2 = 25<BR>\n" );
document.write( "p^2 + (1/p)^2 = 23<BR>\n" );
document.write( "I'll refer to this equation as eq2 for a <font color=blue>substitution</font> step later on.\r<BR>\n" );
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document.write( "Let q = p - (1/p)<BR>\n" );
document.write( "Square both sides to see what happens\r<BR>\n" );
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document.write( "p - (1/p) = q<BR>\n" );
document.write( "[ p - (1/p) ]^2 = q^2<BR>\n" );
document.write( "p*p + p*(-1/p) + (-1/p)*p + (-1/p)(-1/p) = q^2<BR>\n" );
document.write( "p^2 - 1 - 1 + (1/p)^2 = q^2<BR>\n" );
document.write( "p^2 - 2 + (1/p)^2 = q^2<BR>\n" );
document.write( "p^2 + (1/p)^2 - 2 = q^2<BR>\n" );
document.write( "[ p^2 + (1/p)^2 ] - 2 = q^2<BR>\n" );
document.write( "[ <font color=blue size=4>p^2 + (1/p)^2</font> ] - 2 = q^2<BR>\n" );
document.write( "[ <font color=blue size=4>23</font> ] - 2 = q^2 ......... <font color=blue>substitution</font>; use eq2<BR>\n" );
document.write( "21 = q^2<BR>\n" );
document.write( "q^2 = 21<BR>\n" );
document.write( "q = <font color=red>sqrt(21)</font><BR>\n" );
document.write( "p - (1/p) = <font color=red>sqrt(21)</font>\r<BR>\n" );
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document.write( "Another approach would be to solve the equation p + (1/p) = 5 for p\r<BR>\n" );
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document.write( "p + (1/p) = 5<BR>\n" );
document.write( "p * [ p + (1/p) ] = p*5<BR>\n" );
document.write( "p^2 + 1 = 5p<BR>\n" );
document.write( "p^2 - 5p + 1 = 0\r<BR>\n" );
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document.write( "Use the quadratic formula to find these two roots<BR>\n" );
document.write( "p = (5 + sqrt(21))/2<BR>\n" );
document.write( "p = (5 - sqrt(21))/2\r<BR>\n" );
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document.write( "Then use either root to compute p - (1/p)<BR>\n" );
document.write( "I'll let you do these steps.\r<BR>\n" );
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document.write( "Hint:<BR>\n" );
document.write( "If p = (5+sqrt(21))/2, then 1/p = (5 - sqrt(21))/2 after rationalizing the denominator.<BR>\n" );
document.write( "A similar situation happens when p = (5 - sqrt(21))/2<BR>\n" );
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