document.write( "Question 1202304: The sum of the squares of two numbers is 218 and their difference multiplied by the smaller number equals 42. Find the two numbers. \n" ); document.write( "

Algebra.Com's Answer #837033 by ikleyn(52803) $\"\"$ $\"About$

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\n" ); document.write( "The sum of the squares of two numbers is 218 and their difference
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document.write( "First equation is\r\n" );
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document.write( "    x^2 + y^2 = 218.    (1)\r\n" );
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document.write( "Second equation is\r\n" );
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document.write( "    y*(x-y) = 42.       (2)\r\n" );
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document.write( "We do not see an algebra way to solve it; so, will assume the numbers integer\r\n" );
document.write( "and will try to solve the problem using the properties of integer numbers.\r\n" );
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document.write( "Using elementary number theory arguments, from equation (1) it is easy to derive\r\n" );
document.write( "that the numbers x and y should both be odd.\r\n" );
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document.write( "The number 42 has the following odd divisors 1, 3, 7, 21.\r\n" );
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document.write( "So, we write\r\n" );
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document.write( "    x^2 = 218 - y^2\r\n" );
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document.write( "and look, which of these values of y will give us the square of integer x\r\n" );
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document.write( "    218 - 1^2 = 217 is not a square of an integer\r\n" );
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document.write( "    218 - 3^2 = 218 - 9 = 209 is not a square of an integer;\r\n" );
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document.write( "    218 - 7^2 = 218 - 49 = 169 is the square of the integer number 13.\r\n" );
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document.write( "    218 - 21^2 is negative, so it is not a square, at all.\r\n" );
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document.write( "So, only y= 7 is the appropriate candidate, and with x= 13 it really gives a solution.\r\n" );
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\n" ); document.write( "\n" ); document.write( "I wrote this post as an alternative to the solution by @math_tutor2020;
\n" ); document.write( "but now Edwin found the second possible answer, which changed the entire field dramatically.\r
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\n" ); document.write( "\n" ); document.write( "Edwin, my congratulations - your solution is very nice and very unexpected !\r
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document.write( "After seeing the solution by Edwin, I quickly got that there is \r\n" );
document.write( "a straightforward Algebra way to do the same without any tricks.\r\n" );
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document.write( "This way is to express  x =   from equation (2) and substitute it into equation (1)\r\n" );
document.write( "written in the form\r\n" );
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document.write( "     = 218 - .\r\n" );
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document.write( "It will give a biquadratic equation with four roots for y, that will bring four routs for x.\r\n" );
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