document.write( "Question 1209497: Find the range of values of k for which the expression 3 - 4k - (k+3)x - x^2 will be negative for all real values of x.
\n" ); document.write( "[the answer is 3 < k < 7] \n" ); document.write( "

Algebra.Com's Answer #848928 by ikleyn(52781) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "Find the range of values of k for which the expression 3 - 4k - (k+3)x - x^2 will be negative
\n" ); document.write( "for all real values of x.
\n" ); document.write( "[the answer is 3 < k < 7]
\n" ); document.write( "~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " I just solved several similar problems yesterday and today, so at the regular studying process,\r
\n" ); document.write( "\n" ); document.write( " it is just a time for a visitor to turn on his own mind, having several TEMPLATES from me.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "The given quadratic function 3-4k - (k+3)x - x^2  has the leading coefficient -1 at x^2,\r\n" );
document.write( "so it represents a downward parabola.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "In order for this quadratic polynomial be negative at all real values of x, the necessary and \r\n" );
document.write( "sufficient condition is that the discriminant\r\n" );
document.write( "\r\n" );
document.write( "    d = b^2 - 4ac \r\n" );
document.write( "\r\n" );
document.write( "be negative.  Then the square function has no real roots and remains negative for all real values of x.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So, we write the discriminant\r\n" );
document.write( "\r\n" );
document.write( "    d = (-(k+3))^2 - 4*(-1)*(3-4k) = (k+3)^2 + 4*(3-4k) = k^2 + 6k + 9 + 12 - 16k = k^2 - 10k + 21.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "It can be factored\r\n" );
document.write( "\r\n" );
document.write( "    d = (k-7)*(k-3).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So, the discriminant is negative\r\n" );
document.write( "\r\n" );
document.write( "    (k-7)*(k-3) < 0\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "if and only if  the parameter \"k\" is between the roots  3 < k < 7,\r\n" );
document.write( "when the factor (k-7) is negative, while the factor (k-3) is positive.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "At this point, the problem is fully solved.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "ANSWER.  The range of values of k for which the expression  3 - 4k - (k+3)x - x^2  will be negative \r\n" );
document.write( "\r\n" );
document.write( "         for all real values of x  is  3 < k < 7.\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Solved.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );