document.write( "Question 1173404: For what value of b will the line y=-2x+b be tangent to the parabola y=3x^2+4x-1 \n" ); document.write( "

Algebra.Com's Answer #798635 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "For what value of b will the line y=-2x+b be tangent to the parabola y=3x^2+4x-1
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\n" ); document.write( "\n" ); document.write( "            There are TWO WAYS to solve the problem: one way is Algebra, the other way is Calculus.\r
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\n" ); document.write( "\n" ); document.write( "            I will show you both.\r
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\n" ); document.write( "\n" ); document.write( "1.   Calculus way\r
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document.write( "The slope of the given line is -2, the constant value.\r\n" );
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document.write( "The slope of the parabola is  6x+4  at the point with abscissa x.\r\n" );
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document.write( "In order for the line be tangent to the parabola, the necessary condition is \"slope\" = \"slope\",\r\n" );
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document.write( "which gives  -2 = 6x + 4,  and then  x= -1.\r\n" );
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document.write( "At x= -1,  the y-coordinate of the parabola is  3*(-1)^2 + 4*(-1) - 1 = 3*1 - 4 - 1 = -2;\r\n" );
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document.write( "           the y-coordinate of the line     is -2*(-1) + b= 2 + b.\r\n" );
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document.write( "Y-coordinate should be the same, which gives  2 + b = -2, and hence  b= -4.    ANSWER\r\n" );
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\n" ); document.write( "\n" ); document.write( "So, the Calculus solution is completed.\r
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\n" ); document.write( "\n" ); document.write( "2. Algebra way\r
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document.write( "You write this equation\r\n" );
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document.write( "    -2x + b = 3x^2 + 4x-1\r\n" );
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document.write( "to find common point of the line and the parabola.\r\n" );
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document.write( "Its standard form is\r\n" );
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document.write( "    3x^2 + 6x - (1+b) = 0.\r\n" );
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document.write( "The straight line is a tangent to the parabola if and only if there is ONLY ONE common point;\r\n" );
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document.write( "in other words, if there is ONLY ONE solution to the last equation.\r\n" );
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document.write( "It means that the discriminant of this equation is zero\r\n" );
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document.write( "    6^2 + 4*3*(1+b) = 0,\r\n" );
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document.write( "or\r\n" );
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document.write( "    36 + 12*(1+b) = 0\r\n" );
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document.write( "     3 + (1+b) = 0\r\n" );
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document.write( "     b         = -4.\r\n" );
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document.write( "We got the same answer.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Thus the problem is solved and two basic solutions are presented.\r
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