document.write( "Question 1146943: The horizontal line y = k intersects the parabola with equation y = 2(x-3)(x-5) at points A and B. If the length of line segment AB is 6, what is th value of k? \n" ); document.write( "

Algebra.Com's Answer #768420 by ikleyn(52803) $\"\"$ $\"About$

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document.write( "The equation\r\n" );
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document.write( "    2*(x-3)*(x-5) = k\r\n" );
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document.write( "is equivalent to\r\n" );
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document.write( "    2x^2 - 16x + 30 = k,   or\r\n" );
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document.write( "    2x^2 - 16x + (30-k) = 0,  or\r\n" );
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document.write( "     x^2 - 8x +  = 0.\r\n" );
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document.write( "According to Vieta's theorem, the sum of the roots is equal to coefficient at \"x\" with the opposite sign, i.e. 8.\r\n" );
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document.write( "The difference of the roots is equal to 6 (given).\r\n" );
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document.write( "If \"a\" is the greater root, then the other root is (8-a), and the difference between them is\r\n" );
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document.write( "    6 = a - (8-a) = 2a - 8,\r\n" );
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document.write( "which implies  \r\n" );
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document.write( "    2a = 6 + 8 = 14,\r\n" );
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document.write( "     a = 7.\r\n" );
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document.write( "Thus the greater root is  a= 7, while the smaller root is  8-a = 8-7 = 1.\r\n" );
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document.write( "The product of the roots, 7*1, is equal to the constant term  , according to Vieta's theorem (again (!) ).\r\n" );
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document.write( "Thus you have this equation for k\r\n" );
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document.write( "     = 7,\r\n" );
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document.write( "which implies\r\n" );
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document.write( "    30 - k = 14.\r\n" );
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document.write( "Hence,  k = 30-14 = 16.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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\n" ); document.write( "\n" ); document.write( "VERY GOOD problem on Vieta's theorem (!)\r
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