document.write( "Question 1190556: The curve y = ax^2 + bx has gradient 8 when x = 2 and has gradient -10 when x = -1. Find the value of
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Algebra.Com's Answer #822249 by ikleyn(52777) $\"\"$ $\"About$

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document.write( "The problem says that the derivative    is   8  when x= 2,    (1)\r\n" );
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document.write( "                  and the derivative    is -10  when x= -1.   (2)\r\n" );
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document.write( "So, we calculate the derivative   = 2ax + b,\r\n" );
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document.write( "and form two equations aka (1) and (2) for points x= 2  and  x= -1\r\n" );
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document.write( "    2a*2    + b =   8     (3)\r\n" );
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document.write( "    2a*(-1) + b = -10     (4)\r\n" );
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document.write( "Equivalently, this system of equations is\r\n" );
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document.write( "     4a + b =   8         (3')\r\n" );
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document.write( "    -2a + b = -10         (4')\r\n" );
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document.write( "To find \"a\" from these equations, we subtract equation (4') from equation (3'), making Elimination.\r\n" );
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document.write( "We get then\r\n" );
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document.write( "     4a - (-2a) = 8 - (-10)\r\n" );
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document.write( "        6a      =   18\r\n" );
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document.write( "         a      =   18/6 = 3.\r\n" );
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document.write( "Then from equation (3')\r\n" );
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document.write( "     b = 8 - 4a = 8 - 4*3 = 8 - 12 = -4.\r\n" );
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document.write( "ANSWER.  a = 3;  b = -4.\r\n" );
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