document.write( "Question 1086674: If x+y+z=7 and xy+yz+zx=11,then least and largest value of z are ? \n" ); document.write( "
Algebra.Com's Answer #700901 by Edwin McCravy(20056)\"\" \"About 
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If x+y+z=7 and xy+yz+zx=11,then least and largest value of z are ?
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document.write( "Since z=7-(x+y)\r\n" );
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document.write( "z will be smallest when x+y is large as possible, and\r\n" );
document.write( "z will be largest when x+y is small as possible.\r\n" );
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document.write( "So let x+y = s \r\n" );
document.write( "then y = s-x\r\n" );
document.write( "and z = 7-s\r\n" );
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document.write( "z will be smallest when s is large as possible, and\r\n" );
document.write( "z will be largest when s is small as possible.\r\n" );
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document.write( "We substitute in \r\n" );
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document.write( "xy+yz+zx = 11\r\n" );
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document.write( "x(s-x)+(s-x)(7-s)+(7-s)x = 11\r\n" );
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document.write( "sx-x²+7s-s²-7x+sx+7x-sx = 11\r\n" );
document.write( "-x²+sx+7s-s²= 11\r\n" );
document.write( "-x²+sx+7s-s²-11 = 0\r\n" );
document.write( "-x²+sx-s²+7s-11 = 0\r\n" );
document.write( " x²-sx+s²-7s+11 = 0\r\n" );
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document.write( "The discriminant \r\n" );
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document.write( "\"b%5E2-4ac=%28-s%29%5E2-4%281%29%28s%5E2-7s%2B11%29=s%5E2-4s%5E2%2B28s-44=-3s%5E2%2B28s-44\" \r\n" );
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document.write( "must not be negative, so\r\n" );
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document.write( "\"-3s%5E2%2B28s-44%3E=0\"\r\n" );
document.write( "\"3s%5E2-28s%2B44%3C=0\"\r\n" );
document.write( "\"%28s-2%29%283s-22%29%3C=0\"\r\n" );
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document.write( "We solve this inequality:\r\n" );
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document.write( "Critical numbers are s=2 and s=22/3, which are possible\r\n" );
document.write( "values for s.\r\n" );
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document.write( "For the interval \"%28matrix%281%2C3%2C-infinity%2C%22%2C%22%2C2%29%29\"\r\n" );
document.write( "we choose test value 0, the value is +44, so we do not\r\n" );
document.write( "include that interval for s.\r\n" );
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document.write( "For the interval \"%28matrix%281%2C3%2C2%2C%22%2C%22%2C22%2F3%29%29\"\r\n" );
document.write( "we choose test value 3, the value is -13, so we do \r\n" );
document.write( "include that interval for s.\r\n" );
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document.write( "For the interval \"%28matrix%281%2C3%2C22%2F3%2C%22%2C%22%2Cinfinity%29%29\"\r\n" );
document.write( "we choose test value 8, the value is +12, so we do not\r\n" );
document.write( "include that interval for s.\r\n" );
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document.write( "So \"2%3C=s%3C=22%2F3\"\r\n" );
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document.write( "z is smallest when s=22/3, and since z=7-s, the smallest\r\n" );
document.write( "value of z is 7-22/3 = 21/3-22/3 = -1/3\r\n" );
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document.write( "z is largest when s=2, and since z=7-s, the largest\r\n" );
document.write( "value of z is 7-2 = 5.\r\n" );
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document.write( "Edwin
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