document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1052497>Question 1052497</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1)the sum of the ages of a mother and a daughter is 29 year.four years ago the product of their ages in years was 96 years form a quadratic  equation.\r<BR>\n" );
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document.write( "2)the difference of two natural numbers is 3 and difference of their reciprocals is 3/28 form a quadratic  equation.\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #667844 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=667844\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> let the mother's age be x.<BR>\n" );
document.write( "let the daughter's age be y.\r<BR>\n" );
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document.write( "the sum of their ages is 29.\r<BR>\n" );
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document.write( "this mean that x + y = 29.\r<BR>\n" );
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document.write( "4 years ago, the product of their ages was 96.\r<BR>\n" );
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document.write( "the mother's age 4 years ago was x - 4.\r<BR>\n" );
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document.write( "the daughter's age 4 years ago was y - 4.\r<BR>\n" );
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document.write( "the product of their ages 4 years ago was 96.\r<BR>\n" );
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document.write( "this mean that (x-4) * (y-4) = 96.\r<BR>\n" );
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document.write( "in the equation of x + y = 29, solve for y to get:\r<BR>\n" );
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document.write( "y = 29-x.\r<BR>\n" );
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document.write( "in the equation of (x-4) * (y-4) = 96, replace y with 29-x to get:\r<BR>\n" );
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document.write( "(x-4) * (29-x-4) = 96.\r<BR>\n" );
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document.write( "simplify to get:\r<BR>\n" );
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document.write( "(x-4) * (25-x) = 96.\r<BR>\n" );
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document.write( "multiply the factors together to get:\r<BR>\n" );
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document.write( "25x - x^2 - 100 + 4x = 96\r<BR>\n" );
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document.write( "combine like terms to get:\r<BR>\n" );
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document.write( "29x - x^2 - 100 = 96\r<BR>\n" );
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document.write( "subtract 96 from both sides of the equation to get:\r<BR>\n" );
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document.write( "29x - x^2 - 196 = 0\r<BR>\n" );
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document.write( "rearrange the terms in descending order of degree to get:\r<BR>\n" );
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document.write( "-x^2 + 29x - 196 = 0\r<BR>\n" );
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document.write( "that's your quadratic equation in standard form.\r<BR>\n" );
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document.write( "unfortunately, it doesn't factor cleanly, but it does factor and the answer makes sense, even though it doesn't.\r<BR>\n" );
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document.write( "i used a quadratic equation calculator to find the solution.\r<BR>\n" );
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document.write( "the solution to this quadratic equation of -x^2 + 29x - 196 = 0 are:\r<BR>\n" );
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document.write( "x = 18.274917217635 or x = 10.725082782365\r<BR>\n" );
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document.write( "since x + y = 29, then:\r<BR>\n" );
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document.write( "when x = 18.274917217635, y = 10.725082782365\r<BR>\n" );
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document.write( "and when x = 10.725082782365, y = 18.274917217635\r<BR>\n" );
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document.write( "since x is the age of the mother, then x has to be greater than y, so you get:\r<BR>\n" );
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document.write( "age of the mother is 18.274917217635\r<BR>\n" );
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document.write( "age of the daughter is 10.725082782365\r<BR>\n" );
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document.write( "the sum of their ages is 29.\r<BR>\n" );
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document.write( "if you subtract 4 from each of their ages and then multiply those ages together, you get 96.\r<BR>\n" );
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document.write( "the solution is \r<BR>\n" );
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document.write( "since the mother has to be older than the daughter, then let x = 18.274917217635 and y = 10.725082782365.\r<BR>\n" );
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document.write( "not very realistic, but that's how the numbers come out.\r<BR>\n" );
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document.write( "the mother would have had to have the daughter when she was 8.\r<BR>\n" );
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document.write( "not likely.\r<BR>\n" );
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document.write( "add the ages together and you get 29.\r<BR>\n" );
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document.write( "multiply 14.274917217635 * 6.725082782365 and you get 96.\r<BR>\n" );
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document.write( "the requirements of the problem are satisfied.\r<BR>\n" );
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document.write( "therefore the solution looks good.\r<BR>\n" );
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document.write( "that's the part that makes sense.\r<BR>\n" );
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document.write( "the part that doesn't make sense is the mother is only 8 years older than the daughter.\r<BR>\n" );
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document.write( "that's not very likely.\r<BR>\n" );
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document.write( "in fact, it may even be impossible.\r<BR>\n" );
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document.write( "you did not need to go this far however.\r<BR>\n" );
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document.write( "all they wanted was the quadratic equation.\r<BR>\n" );
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document.write( "the solution you are looking for is:\r<BR>\n" );
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document.write( "the standard form of the quadratic equation is -x^2 + 29x - 196 = 0\r<BR>\n" );
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document.write( "NOW TO YOUR SECOND PROBLEM.\r<BR>\n" );
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document.write( "the difference of two natural numbers is 3 and difference of their reciprocals is 3/28 forms a quadratic equation. \r<BR>\n" );
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document.write( "let x equal the first number and y equal the second number.\r<BR>\n" );
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document.write( "these are natural numbers so they have to be positive integers i believe.<BR>\n" );
document.write( "they can sometimes also include 0, depending on who's defining.<BR>\n" );
document.write( "they are definitely not negative.\r<BR>\n" );
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document.write( "if x is the larger number, then you get x - y = 3.\r<BR>\n" );
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document.write( "the difference of their reciprocals is 3/28.\r<BR>\n" );
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document.write( "this means that 1/y - 1/x = 3/28\r<BR>\n" );
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document.write( "note that if x > y, then 1/x < 1/y which makes 1/y - 1/x positive.\r<BR>\n" );
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document.write( "you can use any positive numbers for x and y to confirm this is true.\r<BR>\n" );
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document.write( "so we have two equations.\r<BR>\n" );
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document.write( "x - y = 3\r<BR>\n" );
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document.write( "1/y - 1/x = 3/28\r<BR>\n" );
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document.write( "if we multiply both sides of the second equation by xy, we get:\r<BR>\n" );
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document.write( "x - y = 3xy/28\r<BR>\n" );
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document.write( "if we multiply both sides of this equation by 28, we get:\r<BR>\n" );
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document.write( "28x - 28y = 3xy\r<BR>\n" );
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document.write( "from the equation of x - y = 3, we can solve for x to get:\r<BR>\n" );
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document.write( "x = y+3\r<BR>\n" );
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document.write( "in the equation of 28x - 28y = 3xy, replace x with y+3 to get:\r<BR>\n" );
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document.write( "28(y+3) - 28y = 3(y+3)y\r<BR>\n" );
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document.write( "simplify to get:\r<BR>\n" );
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document.write( "28y + 84 - 28y = 3y^2 + 9y \r<BR>\n" );
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document.write( "combine like terms to get 84 = 3y^2 + 9y\r<BR>\n" );
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document.write( "subtract 84 from both sides of the equation to get 0 = 3y^2 + 9y -84\r<BR>\n" );
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document.write( "flip sides to get 3y^2 + 9y - 84 = 0\r<BR>\n" );
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document.write( "that's your quadratic equation in standard form.\r<BR>\n" );
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document.write( "factor this equation to get (3y-12) * (y+7) = 0\r<BR>\n" );
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document.write( "solve for y to get y = 4 or y = -7\r<BR>\n" );
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document.write( "y has to be a positive integer.\r<BR>\n" );
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document.write( "therefore, y has to be equal to 4.\r<BR>\n" );
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document.write( "let's see if that's correct.\r<BR>\n" );
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document.write( "y is the smaller number.\r<BR>\n" );
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document.write( "if x - y = 3, then x - 4 = 3 which makes x = 7.\r<BR>\n" );
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document.write( "you have x = 7 and y = 4\r<BR>\n" );
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document.write( "7 - 4 = 3\r<BR>\n" );
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document.write( "1/4 - 1/7 should be equal to 3/28.\r<BR>\n" );
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document.write( "common denominator is 28.\r<BR>\n" );
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document.write( "you get 7/28 - 4/28 = 3/28.\r<BR>\n" );
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document.write( "it does equal to 3/28, so the solution looks good.l\r<BR>\n" );
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document.write( "the solution is x = 7 and y = 4.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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