document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=405308>Question 405308</A>: <I><SPAN STYLE=\"word-break: break-all;\"> For which value or values of k, if any, does the quadratic equation in x,<BR>\n" );
document.write( "(k-8)x^2+kx -1 = 0 have exactly one solution? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #286372 by </B><A HREF=/tutors/aboutme.mpl?userid=ankor@dixie-net.com><B>ankor@dixie-net.com(22740)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ankor@dixie-net.com><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=ankor@dixie-net.com><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=286372\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> For which value or values of k, if any, does the quadratic equation in x,<BR>\n" );
document.write( "(k-8)x^2+kx -1 = 0 have exactly one solution?<BR>\n" );
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document.write( "We know that when the discriminant = 0, we have a single value for the roots<BR>\n" );
document.write( "b^2 - 4*a*c = 0<BR>\n" );
document.write( "In this equation a = (k-8), b = k, c = -1<BR>\n" );
document.write( ":<BR>\n" );
document.write( "k^2 - (4*(k-8)*-1) = 0<BR>\n" );
document.write( ":<BR>\n" );
document.write( "k^2 - (-4(k-8))<BR>\n" );
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document.write( "k^2 - (-4k + 32)<BR>\n" );
document.write( "Remove brackets, change the signs<BR>\n" );
document.write( "k^2 + 4k - 32 = 0<BR>\n" );
document.write( "factors to<BR>\n" );
document.write( "(k+8)(k-4) = 0<BR>\n" );
document.write( "two values for k<BR>\n" );
document.write( "k = -8<BR>\n" );
document.write( "k = +4<BR>\n" );
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document.write( "we can prove this<BR>\n" );
document.write( "k=-8<BR>\n" );
document.write( "(-8-8)x^2 - 8x - 1 = 0<BR>\n" );
document.write( "-16x^2 - 8x - 1 = 0<BR>\n" );
document.write( "Multiply by -1<BR>\n" );
document.write( "16x^2 + 8x + 1 = 0<BR>\n" );
document.write( "factors to<BR>\n" );
document.write( "(4x + 1)(4x + 1) = 0<BR>\n" );
document.write( "4x = -1<BR>\n" );
document.write( "x = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2F4 ALIGN=MIDDLE  ALT=\"-1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > a double root with one value<BR>\n" );
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document.write( "and for k = 4<BR>\n" );
document.write( "(4-8)x^2 + 4x - 1 = 0<BR>\n" );
document.write( "-4x^2 + 4x - 1 = 0<BR>\n" );
document.write( "Multiply by -1<BR>\n" );
document.write( "4x^2 - 4x + 1 = 0<BR>\n" );
document.write( "factors to<BR>\n" );
document.write( "(2x - 1)(2x - 1) = 0<BR>\n" );
document.write( "2x = 1<BR>\n" );
document.write( "x = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > a double root with one value<BR>\n" );
document.write( ":<BR>\n" );
document.write( ":<BR>\n" );
document.write( "We can say: k=-8 or k=4 will give one solution for the equation<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );