document.write( "Question 684428: On my last birthday, my friend said to me:
\n" ); document.write( "\"In 15 years' time, your age will be the square of your age 15 years ago!\"
\n" ); document.write( " a) How old am I?
\n" ); document.write( " b) Could I have said:
\n" ); document.write( " \"In 3 years' time, my age will be the square of my age 3 years ago\"? why?
\n" ); document.write( " c)Could I have said:
\n" ); document.write( " \"In 5 years time, my age will be the square of my age 5 years ago\"? why?
\n" ); document.write( " \n" ); document.write( "
Algebra.Com's Answer #424095 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
\"In 15 years' time, your age will be the square of your age 15 years ago!\"
\n" ); document.write( " a) How old am I?
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document.write( "Let your age be A, then\r\n" );
document.write( "Your age in 15 years time = A+15\r\n" );
document.write( "Your age 15 years ago = A-15\r\n" );
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document.write( "A + 15 = (A - 15)²\r\n" );
document.write( "A + 15 = (A - 15)(A - 15)\r\n" );
document.write( "A + 15 = A² - 30A + 225\r\n" );
document.write( "     0 = A² - 31A + 210\r\n" );
document.write( "     0 = (A - 21)(A - 10)\r\n" );
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document.write( "         A - 21 = 0;   A - 10 = 0\r\n" );
document.write( "              A = 21;       A = 10\r\n" );
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document.write( "Checking A = 21.  in 16 years you will be 36, and 15 years ago you\r\n" );
document.write( "wer 6, and 36 is the square of 6.\r\n" );
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document.write( "Checking A = 10.  in 15 years you will be 25, and 15 years ago you\r\n" );
document.write( "lacked 5 years being born, so you were -5, and 25 is the square of -5.\r\n" );
document.write( "However we don't normally consider yet unborn people to have negative\r\n" );
document.write( "ages, so we discard A = 10 and say the only solution is A = 21. \r\n" );
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\n" ); document.write( "b) Could I have said:
\n" ); document.write( " \"In 3 years' time, my age will be the square of my age 3 years ago\"? why?
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document.write( "Let's find out:\r\n" );
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document.write( "Let your age be A, then\r\n" );
document.write( "Your age in 3 years time = A+3\r\n" );
document.write( "Your age 15 years ago = A-3\r\n" );
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document.write( "A + 3 = (A - 3)²\r\n" );
document.write( "A + 3 = (A - 3)(A - 3)\r\n" );
document.write( "A + 3 = A² - 6A + 9\r\n" );
document.write( "    0 = A² - 7A + 6\r\n" );
document.write( "    0 = (A - 6)(A - 1)\r\n" );
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document.write( "         A - 6 = 0;   A - 1 = 0\r\n" );
document.write( "             A = 6;       A = 1\r\n" );
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document.write( "Checking A = 6.  in 3 years you will be 9, and 3 years ago you\r\n" );
document.write( "were 3, and 9 is the square of 3.\r\n" );
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document.write( "Checking A = 1.  in 3 years you will be 4, and 4 years ago you\r\n" );
document.write( "lacked 2 years being born, so you were -2, and 4 is the square of -2.\r\n" );
document.write( "However we don't normally consider yet unborn people to have negative\r\n" );
document.write( "ages, so we discard A = 1 and say the only solution is A = 6.\r\n" );
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document.write( "So the answer is \"yes\", because age 6 checks. \r\n" );
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\n" ); document.write( "b) Could I have said:
\n" ); document.write( " \"In 5 years' time, my age will be the square of my age 5 years ago\"? why?
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\r\n" );
document.write( "Let's find out:\r\n" );
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document.write( "Let your age be A, then\r\n" );
document.write( "Your age in 5 years time = A+5\r\n" );
document.write( "Your age 5 years ago = A-5\r\n" );
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document.write( "A + 5 = (A - 5)²\r\n" );
document.write( "A + 5 = (A - 5)(A - 5)\r\n" );
document.write( "A + 5 = A² - 10A + 25\r\n" );
document.write( "    0 = A² - 11A + 20\r\n" );
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document.write( "Unlike the others, this doesn't factor.  So we suspect\r\n" );
document.write( "there can be no solution unless we allow fractions of\r\n" );
document.write( "years.   We use the quadratic formula:\r\n" );
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document.write( "x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}} \r\n" );
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document.write( "x = (-(-11) +- sqrt((-11)^2-4*(1)*(20) ))/(2*(1)) }}}\r\n" );
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document.write( "x = (11 +- sqrt((121-80 ))/2 }}}\r\n" );
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document.write( "x = (11 +- sqrt((41 ))/2 }}}\r\n" );
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document.write( "The two solutions are approximately 8.7 and 2.3.\r\n" );
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document.write( "Checking A = 8.7, approximately.  in 5 years you will be approximately\r\n" );
document.write( "13.7, and 5 years ago you were approximately 3.7, and 13.7 is approximately\r\n" );
document.write( "the square of 3.7 (The exact square of 3.7 is 13.69, so it approzimately\r\n" );
document.write( "works).  However we do not normally use fractions of years in ages, so\r\n" );
document.write( "we should say the answer is \"no, because there is no whole number age\"\r\n" );
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document.write( "You weren't asked for this, but in general:\r\n" );
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document.write( "If I am n(n+1)/2 years old then in n(n-1)/2 years' time I will be \r\n" );
document.write( "the square of my age n(n-1)/2 years ago\".\r\n" );
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document.write( "Edwin
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