document.write( "Question 1209488: Find the range of values of k for which the expression x^2 + kx + (k+3) is positive for all real values of x. Hence, fine the range of values of x for which (2x-1)(3-x)/x^2+4x+7 < 0 \n" ); document.write( "

Algebra.Com's Answer #848924 by ikleyn(52810) $\"\"$ $\"About$

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\n" ); document.write( "(a) Find the range of values of k for which the expression x^2 + kx + (k+3) is positive for all real values of x.
\n" ); document.write( "(b) Hence, $\"highlight%28cross%28fine%29%29\"$ find the range of values of x for which (2x-1)(3-x)/(x^2+4x+7) < 0
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\n" ); document.write( "\n" ); document.write( "        In this assignment, there are two tasks: (a) and (b).\r
\n" ); document.write( "\n" ); document.write( "        I will solve them separately to avoid mess.\r
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\n" ); document.write( "\n" ); document.write( "        Also notice that I edited your inequality in part (b) according to common sense.\r
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document.write( "                Part (a)\r\n" );
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document.write( "Consider the discriminant of the quadratic polynomial x^2 + kx + (k+3).\r\n" );
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document.write( "The discriminant is  d = b^2 - 4ac = k^2 - 4*(k+3) = k^2 - 4k - 12 = (k-6)*(k+2).\r\n" );
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document.write( "We see that the discriminant is negative in the interval -2 < k < 6.\r\n" );
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document.write( "It means that the polynomial does not have real zeroes if -2 < k < 6.\r\n" );
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document.write( "From the other side hand, its leading coefficient at x^2, \"1\", is positive. \r\n" );
document.write( "It means that the polynomial  x^2 + kx + (k+3) is always positive, for all real values of x,\r\n" );
document.write( "if k is in the open interval (-2,6).\r\n" );
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document.write( "Thus, part (a) is solved/answered completely.\r\n" );
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document.write( "                Part (b)\r\n" );
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document.write( "Consider this rational function  .\r\n" );
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document.write( "Its denominator is of the form x^2 + kx + (k+3) at k = 4.\r\n" );
document.write( "We considered such polynomials in part (a) and proved that for k from the interval (-2,6) the polynomial is always positive,\r\n" );
document.write( "for all real values of x.  The value of k= 4 is from this interval - so, the polynomial x^2 + 4x + 7 in the denominator\r\n" );
document.write( "is always positive, for all real values of x.\r\n" );
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document.write( "Therefore, inequality \r\n" );
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document.write( "     < 0     (1)\r\n" );
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document.write( "is equivalent to this simplified inequality\r\n" );
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document.write( "    (2x-1)*(3-x) < 0.    (2)\r\n" );
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document.write( "This inequality (2) has the leading coefficient -2 at x^2 and the roots 1/2 and 3,\r\n" );
document.write( "so the left side is the downward parabola with x-intercepts 1/2 and 3.\r\n" );
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document.write( "So, the inequality (2) has the solution set   < x < 3.\r\n" );
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document.write( "It implies that inequality (1) has the same solution set   < x < 3.\r\n" );
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document.write( "Thus the range of values of x for which   < 0  is  (,).    ANSWER\r\n" );
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document.write( "At this point, part (b) is solved completely.\r\n" );
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\n" ); document.write( "\n" ); document.write( "The solution is complete and all questions are answered.\r
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