document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209534>Question 1209534</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 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.<BR>\n" );
document.write( "[the answer is 3 < k < 7] </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849293 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849293\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to find the range of values for k:\r<BR>\n" );
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document.write( "**1. Rewrite the expression:**\r<BR>\n" );
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document.write( "The given expression is 3 - 4k - (k+3)x - x². We can rewrite this as:\r<BR>\n" );
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document.write( "-x² - (k+3)x + (3 - 4k)\r<BR>\n" );
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document.write( "**2. Consider the quadratic form:**\r<BR>\n" );
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document.write( "This is a quadratic expression in x of the form ax² + bx + c, where:\r<BR>\n" );
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document.write( "* a = -1<BR>\n" );
document.write( "* b = -(k+3)<BR>\n" );
document.write( "* c = (3 - 4k)\r<BR>\n" );
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document.write( "**3. Condition for negativity:**\r<BR>\n" );
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document.write( "For the quadratic expression to be negative for all real values of x, two conditions must be met:\r<BR>\n" );
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document.write( "* **The parabola must open downwards:** This is already satisfied since a = -1 (negative).<BR>\n" );
document.write( "* **The quadratic must have no real roots:**  This means the discriminant (b² - 4ac) must be less than zero.\r<BR>\n" );
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document.write( "**4. Apply the discriminant condition:**\r<BR>\n" );
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document.write( "The discriminant is:\r<BR>\n" );
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document.write( "b² - 4ac = [-(k+3)]² - 4*(-1)*(3 - 4k)<BR>\n" );
document.write( "        = (k² + 6k + 9) + 4(3 - 4k)<BR>\n" );
document.write( "        = k² + 6k + 9 + 12 - 16k<BR>\n" );
document.write( "        = k² - 10k + 21\r<BR>\n" );
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document.write( "For no real roots, the discriminant must be less than zero:\r<BR>\n" );
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document.write( "k² - 10k + 21 < 0\r<BR>\n" );
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document.write( "**5. Solve the inequality:**\r<BR>\n" );
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document.write( "Factor the quadratic:\r<BR>\n" );
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document.write( "(k - 3)(k - 7) < 0\r<BR>\n" );
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document.write( "This inequality is satisfied when k is between the roots 3 and 7.\r<BR>\n" );
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document.write( "**6. Final answer:**\r<BR>\n" );
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document.write( "Therefore, the expression 3 - 4k - (k+3)x - x² will be negative for all real values of x when:\r<BR>\n" );
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document.write( "3 < k < 7<BR>\n" );
document.write( " </SPAN>\n" );
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