document.write( "Question 1162180: determine k such that g(x)=3x^2+8x+k intersects f(x)=2x-1 at ine point. \n" ); document.write( "

Algebra.Com's Answer #785916 by ikleyn(52781) $\"\"$ $\"About$

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document.write( "They want you find the value of \"k\" in a way that the straight line f(x) = 2x-1 be a tangent line \r\n" );
document.write( "to the parabola  g(x) = 3x^2 + 8x + k.\r\n" );
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document.write( "It means that the equation\r\n" );
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document.write( "    3x^2 + 8x + k = 2x - 1 \r\n" );
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document.write( "has only one real root (the two roots merge into one root).\r\n" );
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document.write( "The equation is equivalent to\r\n" );
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document.write( "    3x^2 + 6x + (k+1) = 0.     (*)\r\n" );
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document.write( "Its discriminant is\r\n" );
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document.write( "    b^2 - 4ac =  = 36 - 12(k+1)\r\n" );
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document.write( "We want the discriminant be zero\r\n" );
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document.write( "    36 - 12(k+1) = 0,   or   36 = 12(k+1),   k+1 = 36/12 = 3,  k = 3-1 = 2.\r\n" );
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document.write( "ANSWER.  k = 2.\r\n" );
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document.write( "CHECK.  At k= 2, the equation (*)  becomes  3x^3 +6x + 3 = 0.\r\n" );
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document.write( "        It is equivalent to  x^2 + 2x + 1 = 0,  which  factors into   = 0  and has only one real solution x= -1.\r\n" );
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document.write( "        The point  (x,f(x)) = (-1,f(-1)) = (-1,-3)  is the same as  the point  (x,g(x)) = (-1,g(-1)) = (-1,-3).\r\n" );
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document.write( "        It is the unique common point of the two given lines.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved. // The problem was solved using Algebra only, without using Calculus.\r
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