document.write( "Question 1188448: Determine the restrictions on the y-intercept so that y = 3x^2 + 6x - 1 intersects with a line with slope 2 in more than one place. \n" ); document.write( "

Algebra.Com's Answer #819590 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "Determine the restrictions on the y-intercept so that y = 3x^2 + 6x - 1 intersects
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\n" ); document.write( "\n" ); document.write( " The condition in the post is over-twisted, \r
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document.write( "            Determine the restrictions on the y-intercept of the straight line of the slope 2\r\n" );
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document.write( "            so it intersect the curve y = 3x^2 + 6x - 1 in more than one point.\r\n" );
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\n" ); document.write( "\n" ); document.write( " SOLUTION\r
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document.write( "The straight line equation is  y = 2x + k,  where \"k\" is the y-interception value.\r\n" );
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document.write( "We should determine this value of \"k\" from the condition that the equation\r\n" );
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document.write( "    3x^2 + 6x - 1 = 2x + k\r\n" );
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document.write( "has more than one solution.\r\n" );
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document.write( "We rewrite this equation in standard quadratic form\r\n" );
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document.write( "    3x^2 + 4x - (k+1) = 0.\r\n" );
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document.write( "The condition that it has more than one real solution means that the discriminant is positive\r\n" );
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document.write( "    d = b^2 - 4ac > 0,  where  a= 3;  b= 4  and  c= -(k+1).\r\n" );
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document.write( "So, the discriminant is  \r\n" );
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document.write( "    d = 16 + 4*3*(k+1) = 16 + 12(k+1).\r\n" );
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document.write( "The condition d > o is\r\n" );
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document.write( "    16 + 12(k+1) > 0\r\n" );
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document.write( "or\r\n" );
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document.write( "    12(k+1) > -16\r\n" );
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document.write( "       k+1 > -  = - \r\n" );
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document.write( "         k > -  - 1 = - .\r\n" );
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document.write( "ANSWER.  The y-intercept \"k\" must be greater than  -  = -2  :  k > -2 .\r\n" );
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