document.write( "Question 973230: Find three numbers in continued proportion such that their sum is 14 and sum
\n" ); document.write( "of their squares is 84. \n" ); document.write( "
Algebra.Com's Answer #595485 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
Find three numbers in continued proportion such that their sum is 14 and sum
\n" ); document.write( "of their squares is 84.
\n" ); document.write( "
\r\n" );
document.write( "\"system%28%22x%3Ay%22+=+%22y%3Az%22%2C+x%2By%2Bz+=+14%2C+x%5E2%2By%5E2%2Bz%5E2+=+84%29\"\r\n" );
document.write( "\r\n" );
document.write( "\"system%28x%2Fy+=+y%2Fz%2C+x%2By%2Bz+=+14%2C+x%5E2%2By%5E2%2Bz%5E2+=+84%29\"\r\n" );
document.write( "\r\n" );
document.write( "\"system%28xz+=+y%5E2%2C+x%2By%2Bz+=+14%2C+x%5E2%2By%5E2%2Bz%5E2+=+84%29\"\r\n" );
document.write( "                           __\r\n" );
document.write( "Substitute xz for y² and ±√xz for y in 3rd and 2nd equations:\r\n" );
document.write( "\r\n" );
document.write( "\"system%28x+%2B-+sqrt%28xz%29%2B+z+=+14%2C+x%5E2%2Bxz%2Bz%5E2+=+84%29\"\r\n" );
document.write( "                         __\r\n" );
document.write( "Solve 1st equation for ±√xy\r\n" );
document.write( "\r\n" );
document.write( "\"system%28%22%22+%2B-+sqrt%28xz%29=+14-x-z%2C+x%5E2%2Bxz%2Bz%5E2+=+84%29\"\r\n" );
document.write( "\r\n" );
document.write( "Square both sides of 1st equation\r\n" );
document.write( "\r\n" );
document.write( "\"system%28xz=+196%2Bx%5E2%2Bz%5E2-28x-28z%2B2xz%2C+x%5E2%2Bxz%2Bz%5E2+=+84%29\"\r\n" );
document.write( "\r\n" );
document.write( "Get -x²-xz-z² on the left of the 1st equation:\r\n" );
document.write( "\r\n" );
document.write( "\"system%28-x%5E2-xz-z%5E2=196-28x-28z%2C+x%5E2%2Bxz%2Bz%5E2+=+84%29\"\r\n" );
document.write( "\r\n" );
document.write( "Add the two equations\r\n" );
document.write( "\r\n" );
document.write( "\"0=280-28x-28z\"\r\n" );
document.write( "\r\n" );
document.write( "Divide through by 28\r\n" );
document.write( "\r\n" );
document.write( "\"0=10-x-z\"\r\n" );
document.write( "\r\n" );
document.write( "\"x%2Bz=10\"\r\n" );
document.write( "\r\n" );
document.write( "Put that with original equation x+y+z = 14\r\n" );
document.write( "\r\n" );
document.write( "\"system%28x%2By%2Bz=14%2Cx%2Bz=10%29\"\r\n" );
document.write( "\r\n" );
document.write( "Subtract the second equation from the first:\r\n" );
document.write( "\r\n" );
document.write( "y=4, and since x+z=10, z=10-x\r\n" );
document.write( "\r\n" );
document.write( "Substitute those in the original equation x²+y²+z²=84 \r\n" );
document.write( "\r\n" );
document.write( "        x²+y²+z² = 84\r\n" );
document.write( "   x²+4²+(10-x)² = 84\r\n" );
document.write( "x²+16+100-20x+x² = 84\r\n" );
document.write( "     2x²-20x+116 = 84\r\n" );
document.write( "      2x²-20x+32 = 0\r\n" );
document.write( "\r\n" );
document.write( "Divide through by 2\r\n" );
document.write( "\r\n" );
document.write( "       x²-10x+16 = 0\r\n" );
document.write( "      (x-8)(x-2) = 0\r\n" );
document.write( "   x-8 = 0;    x-2 = 0\r\n" );
document.write( "     x = 8;      x = 2\r\n" );
document.write( "     z = 10-x;   z = 10-x\r\n" );
document.write( "     z = 10-8;   z = 10-2\r\n" );
document.write( "     z = 2;      z = 8\r\n" );
document.write( "\r\n" );
document.write( "So there seems to be two solutions:\r\n" );
document.write( "\r\n" );
document.write( "(x,y,z) = (8,4,2)  \r\n" );
document.write( "(x,y,z) = (2,4,8)\r\n" );
document.write( "\r\n" );
document.write( "Checking: 8:4 = 4:2  that checks\r\n" );
document.write( "          2:4 = 4:8  that checks\r\n" );
document.write( "\r\n" );
document.write( "Actually there is really only 1 solution because\r\n" );
document.write( "\r\n" );
document.write( "if x:y = y:z, then z:y = y:x and vice-versa.\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );