document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=109585>Question 109585</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I am supposed to find k in the following quadratic equation:\r<BR>\n" );
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document.write( "(k+3)x^2 + 2kx + 4 = 0.\r<BR>\n" );
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document.write( "I have tried using the quadratic formula and unless I'm doing something wrong I don't come up with k. I have tried using the discriminant to find k with no success. I must be missing something easy.\r<BR>\n" );
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document.write( "Thanks for your help, Don </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #79868 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=79868\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Are you trying to find a k that yields one real solution? If you are, then the discriminant must equal zero. So \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac=0 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > which means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282k%29%5E2-4%28k%2B3%29%284%29=0 ALIGN=MIDDLE  ALT=\"%282k%29%5E2-4%28k%2B3%29%284%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282k%29%5E2-4%28k%2B3%29%284%29=0 ALIGN=MIDDLE  ALT=\"%282k%29%5E2-4%28k%2B3%29%284%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4k%5E2-16k-48=0 ALIGN=MIDDLE  ALT=\"4k%5E2-16k-48=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\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( "  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=4%2Ak%5E2-16%2Ak-48=0 ALIGN=MIDDLE  ALT=\"4%2Ak%5E2-16%2Ak-48=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ( notice <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=4 ALIGN=MIDDLE  ALT=\"a=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-16 ALIGN=MIDDLE  ALT=\"b=-16\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=-48 ALIGN=MIDDLE  ALT=\"c=-48\"   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--16+%2B-+sqrt%28+%28-16%29%5E2-4%2A4%2A-48+%29%29%2F%282%2A4%29 ALIGN=MIDDLE  ALT=\"k+=+%28--16+%2B-+sqrt%28+%28-16%29%5E2-4%2A4%2A-48+%29%29%2F%282%2A4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in a=4, b=-16, and c=-48<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+%2816+%2B-+sqrt%28+%28-16%29%5E2-4%2A4%2A-48+%29%29%2F%282%2A4%29 ALIGN=MIDDLE  ALT=\"k+=+%2816+%2B-+sqrt%28+%28-16%29%5E2-4%2A4%2A-48+%29%29%2F%282%2A4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Negate -16 to get 16<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+%2816+%2B-+sqrt%28+256-4%2A4%2A-48+%29%29%2F%282%2A4%29 ALIGN=MIDDLE  ALT=\"k+=+%2816+%2B-+sqrt%28+256-4%2A4%2A-48+%29%29%2F%282%2A4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square -16 to get 256  (note: remember when you square -16, you must square the negative as well. This is because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-16%29%5E2=-16%2A-16=256 ALIGN=MIDDLE  ALT=\"%28-16%29%5E2=-16%2A-16=256\"   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+=+%2816+%2B-+sqrt%28+256%2B768+%29%29%2F%282%2A4%29 ALIGN=MIDDLE  ALT=\"k+=+%2816+%2B-+sqrt%28+256%2B768+%29%29%2F%282%2A4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2A-48%2A4 ALIGN=MIDDLE  ALT=\"-4%2A-48%2A4\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=768 ALIGN=MIDDLE  ALT=\"768\"   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+=+%2816+%2B-+sqrt%28+1024+%29%29%2F%282%2A4%29 ALIGN=MIDDLE  ALT=\"k+=+%2816+%2B-+sqrt%28+1024+%29%29%2F%282%2A4%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+=+%2816+%2B-+32%29%2F%282%2A4%29 ALIGN=MIDDLE  ALT=\"k+=+%2816+%2B-+32%29%2F%282%2A4%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+=+%2816+%2B-+32%29%2F8 ALIGN=MIDDLE  ALT=\"k+=+%2816+%2B-+32%29%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply 2 and 4 to get 8<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+=+%2816+%2B+32%29%2F8 ALIGN=MIDDLE  ALT=\"k+=+%2816+%2B+32%29%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+%2816+-+32%29%2F8 ALIGN=MIDDLE  ALT=\"k+=+%2816+-+32%29%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  Lets look at the first part:<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%2816+%2B+32%29%2F8 ALIGN=MIDDLE  ALT=\"x=%2816+%2B+32%29%2F8\"   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=48%2F8 ALIGN=MIDDLE  ALT=\"k=48%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add the terms in the numerator<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=6 ALIGN=MIDDLE  ALT=\"k=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide<BR>\n" );
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document.write( "  So one answer is<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=6 ALIGN=MIDDLE  ALT=\"k=6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  Now lets look at the second part:<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%2816+-+32%29%2F8 ALIGN=MIDDLE  ALT=\"x=%2816+-+32%29%2F8\"   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=-16%2F8 ALIGN=MIDDLE  ALT=\"k=-16%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract the terms in the numerator<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-2 ALIGN=MIDDLE  ALT=\"k=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide<BR>\n" );
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document.write( "  So another answer is<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-2 ALIGN=MIDDLE  ALT=\"k=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  So our solutions are:<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=6 ALIGN=MIDDLE  ALT=\"k=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-2 ALIGN=MIDDLE  ALT=\"k=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "So if k=-2 or k=6, then it will give us one solution for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28k%2B3%29x%5E2+%2B+2kx+%2B+4+ ALIGN=MIDDLE  ALT=\"%28k%2B3%29x%5E2+%2B+2kx+%2B+4+\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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