document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=836283>Question 836283</A>: <I><SPAN STYLE=\"word-break: break-all;\"> For what value(s) of k will the equation kx^2+(k+1)x-1=0 have exactly one solution?<BR>\n" );
document.write( "My teacher has never taught us this. Please help! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #504074 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=504074\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ax^2 + bx + c = 0 will only have one solution if and only if the discriminant D is 0, so D = 0\r<BR>\n" );
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document.write( "D = b^2 - 4ac\r<BR>\n" );
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document.write( "0 = b^2 - 4ac\r<BR>\n" );
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document.write( "0 = (k+1)^2 - 4(k)(-1)\r<BR>\n" );
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document.write( "0 = (k+1)^2 + 4k\r<BR>\n" );
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document.write( "0 = k^2 + 2k + 1 + 4k\r<BR>\n" );
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document.write( "0 = k^2 + 6k + 1\r<BR>\n" );
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document.write( "k^2 + 6k + 1 = 0\r<BR>\n" );
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document.write( "Now use the quadratic formula to solve for k\r<BR>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=quadratic_formula.solver>Quadratic Formula</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Let's use the quadratic formula to solve for k:<BR><BR>\n" );
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document.write( "  Starting with the general quadratic<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ak%5E2%2Bbk%2Bc=0 ALIGN=MIDDLE  ALT=\"ak%5E2%2Bbk%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  the general solution using the quadratic equation is:<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29 ALIGN=MIDDLE  ALT=\"k+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  So lets solve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k%5E2%2B6%2Ak%2B1=0 ALIGN=MIDDLE  ALT=\"k%5E2%2B6%2Ak%2B1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ( notice <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=1 ALIGN=MIDDLE  ALT=\"a=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=6 ALIGN=MIDDLE  ALT=\"b=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=1 ALIGN=MIDDLE  ALT=\"c=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+%28-6+%2B-+sqrt%28+%286%29%5E2-4%2A1%2A1+%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"k+=+%28-6+%2B-+sqrt%28+%286%29%5E2-4%2A1%2A1+%29%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in a=1, b=6, and c=1<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+%28-6+%2B-+sqrt%28+36-4%2A1%2A1+%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"k+=+%28-6+%2B-+sqrt%28+36-4%2A1%2A1+%29%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square 6 to get 36  <BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+%28-6+%2B-+sqrt%28+36%2B-4+%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"k+=+%28-6+%2B-+sqrt%28+36%2B-4+%29%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2A1%2A1 ALIGN=MIDDLE  ALT=\"-4%2A1%2A1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4 ALIGN=MIDDLE  ALT=\"-4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+%28-6+%2B-+sqrt%28+32+%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"k+=+%28-6+%2B-+sqrt%28+32+%29%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms in the radicand (everything under the square root)<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+%28-6+%2B-+4%2Asqrt%282%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"k+=+%28-6+%2B-+4%2Asqrt%282%29%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify the square root (note: If you need help with simplifying the square root, check out this <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Radicals/simplifying-square-roots.solver> solver</a>)<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+%28-6+%2B-+4%2Asqrt%282%29%29%2F2 ALIGN=MIDDLE  ALT=\"k+=+%28-6+%2B-+4%2Asqrt%282%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply 2 and 1 to get 2<BR>\n" );
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document.write( "  So now the expression breaks down into two parts<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+%28-6+%2B+4%2Asqrt%282%29%29%2F2 ALIGN=MIDDLE  ALT=\"k+=+%28-6+%2B+4%2Asqrt%282%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+%28-6+-+4%2Asqrt%282%29%29%2F2 ALIGN=MIDDLE  ALT=\"k+=+%28-6+-+4%2Asqrt%282%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  Now break up the fraction<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-6%2F2%2B4%2Asqrt%282%29%2F2 ALIGN=MIDDLE  ALT=\"k=-6%2F2%2B4%2Asqrt%282%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-6%2F2-4%2Asqrt%282%29%2F2 ALIGN=MIDDLE  ALT=\"k=-6%2F2-4%2Asqrt%282%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  Simplify<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-3%2B2%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"k=-3%2B2%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-3-2%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"k=-3-2%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  So the solutions are:<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-3%2B2%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"k=-3%2B2%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-3-2%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"k=-3-2%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "=========================================================================<BR>\n" );
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document.write( "Final Answer: \r<BR>\n" );
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document.write( "The two solutions are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+-3%2B2%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"k+=+-3%2B2%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+-3-2%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"k+=+-3-2%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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