document.write( "Question 764499: the function f(x) = x^2 + 6x +20 + k(x^2 -3x -12) where k is real
\n" ); document.write( "find the value of k and the value of p given that the minimum value of f(x) is p and that f(-2) =p
\n" ); document.write( "the answers are 2/7 anf 80/7\r
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Algebra.Com's Answer #465504 by Edwin McCravy(20055) $\"\"$ $\"About$

You can put this solution on YOUR website!
the function f(x) = x^2 + 6x +20 + k(x^2 -3x -12) where k is real
\n" ); document.write( "find the value of k and the value of p given that the minimum value of f(x) is p and that f(-2) =p
\n" ); document.write( "the answers are 2/7 and 80/7\r
\n" ); document.write( "\n" ); document.write( "i am happy if you just explain to me what to do, even briefly.
\n" ); document.write( "

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document.write( "f(x) = x² + 6x + 20 + k(x² - 3x - 12)\r\n" );
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document.write( "f(x) = x² + 6x + 20 + kx² - 3kx - 12k\r\n" );
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document.write( "f(x) = x² + kx² + 6x - 3kx + 20 - 12k\r\n" );
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document.write( "f(x) = (1 + k)x² + (6 - 3k)x + 20 - 12k\r\n" );
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document.write( "We are told that f(-2) = p, so\r\n" );
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document.write( "f(-2) = (1 + k)(-2)² + (6 - 3k)(-2) + 20 - 12k = p\r\n" );
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document.write( "              (1 + k)(4) - 12  + 6k + 20 - 12k = p\r\n" );
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document.write( "                               4 + 4k + 8 - 6k = p\r\n" );
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document.write( "                                       12 - 2k = p\r\n" );
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document.write( "The minimum value of f(x) = ax² + bx + c is \r\n" );
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document.write( "f(x) = (1 + k)x² + (6 - 3k)x + (20 - 12k)\r\n" );
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document.write( " = \r\n" );
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document.write( "                       \r\n" );
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document.write( "                       \r\n" );
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document.write( "We are told this minimum value is equal to p, and we have above that\r\n" );
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document.write( "12 - 2k = p, so our equation is\r\n" );
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document.write( "Multiply through by the LCD of 4(1 + k)\r\n" );
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document.write( "(6 - 3k)² - 2(6 - 3k)² = (-8 + 10k)4(1 + k)\r\n" );
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document.write( "            -(6 - 3k)² = 4(-8 + 10k)(1 + k)\r\n" );
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document.write( "     -(36 - 36k + 9k²) = 4(-8 + 2k + 10k²)\r\n" );
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document.write( "       -36 + 36k - 9k² = -32 + 8k + 40k²\r\n" );
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document.write( "       -49k² + 28k - 4 = 0\r\n" );
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document.write( "        49k² - 28k + 4 = 0\r\n" );
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document.write( "             (7k - 2)² = 0\r\n" );
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document.write( "                7k - 2 = 0\r\n" );
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document.write( "                    7k = 2\r\n" );
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document.write( "                     k =  \r\n" );
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document.write( "Then since 12 - 2k = p \r\n" );
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document.write( "                     p = 12 - 2k \r\n" );
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document.write( "                     p = 12 - 2\r\n" );
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document.write( "                     p = \r\n" );
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document.write( "                     p =  \r\n" );
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document.write( "Edwin

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