document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=98447>Question 98447</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Most quadratic eqautions Ive come across so far have given the problem as something equals 0 format. This one below is the first Ive seen in this format so its got me scratching my head. <BR>\n" );
document.write( " Use the quadratic equation to solve: 6n^2= -8n-1 <BR>\n" );
document.write( "Give the answer in decimal form.<BR>\n" );
document.write( "This neither has the equals 0 as a start point or ask me what to solve for. It just says solve. Help please.<BR>\n" );
document.write( "Thank you. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #71617 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=71617\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Well if it's not in the \"equals 0\" format, you must get all terms to one side\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6n%5E2=+-8n-1 ALIGN=MIDDLE  ALT=\"6n%5E2=+-8n-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given quadratic\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=+-6n%5E2-8n-1 ALIGN=MIDDLE  ALT=\"0=+-6n%5E2-8n-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6n%5E2 ALIGN=MIDDLE  ALT=\"6n%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides to get all terms to one side. Now one side equals zero\r<BR>\n" );
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document.write( "Let's use the quadratic formula to solve for n:\r<BR>\n" );
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document.write( "Starting with the general quadratic\r<BR>\n" );
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document.write( "the general solution using the quadratic equation is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29 ALIGN=MIDDLE  ALT=\"n+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So lets solve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6%2An%5E2-8%2An-1=0 ALIGN=MIDDLE  ALT=\"-6%2An%5E2-8%2An-1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ( notice <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=-6 ALIGN=MIDDLE  ALT=\"a=-6\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-8 ALIGN=MIDDLE  ALT=\"b=-8\"   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;\" >)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+%28--8+%2B-+sqrt%28+%28-8%29%5E2-4%2A-6%2A-1+%29%29%2F%282%2A-6%29 ALIGN=MIDDLE  ALT=\"n+=+%28--8+%2B-+sqrt%28+%28-8%29%5E2-4%2A-6%2A-1+%29%29%2F%282%2A-6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in a=-6, b=-8, and c=-1\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+%288+%2B-+sqrt%28+%28-8%29%5E2-4%2A-6%2A-1+%29%29%2F%282%2A-6%29 ALIGN=MIDDLE  ALT=\"n+=+%288+%2B-+sqrt%28+%28-8%29%5E2-4%2A-6%2A-1+%29%29%2F%282%2A-6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Negate -8 to get 8\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+%288+%2B-+sqrt%28+64-4%2A-6%2A-1+%29%29%2F%282%2A-6%29 ALIGN=MIDDLE  ALT=\"n+=+%288+%2B-+sqrt%28+64-4%2A-6%2A-1+%29%29%2F%282%2A-6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square -8 to get 64  (note: remember when you square -8, you must square the negative as well. This is because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-8%29%5E2=-8%2A-8=64 ALIGN=MIDDLE  ALT=\"%28-8%29%5E2=-8%2A-8=64\"   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=n+=+%288+%2B-+sqrt%28+64%2B-24+%29%29%2F%282%2A-6%29 ALIGN=MIDDLE  ALT=\"n+=+%288+%2B-+sqrt%28+64%2B-24+%29%29%2F%282%2A-6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2A-1%2A-6 ALIGN=MIDDLE  ALT=\"-4%2A-1%2A-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-24 ALIGN=MIDDLE  ALT=\"-24\"   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=n+=+%288+%2B-+sqrt%28+40+%29%29%2F%282%2A-6%29 ALIGN=MIDDLE  ALT=\"n+=+%288+%2B-+sqrt%28+40+%29%29%2F%282%2A-6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms in the radicand (everything under the square root)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+%288+%2B-+2%2Asqrt%2810%29%29%2F%282%2A-6%29 ALIGN=MIDDLE  ALT=\"n+=+%288+%2B-+2%2Asqrt%2810%29%29%2F%282%2A-6%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>)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+%288+%2B-+2%2Asqrt%2810%29%29%2F-12 ALIGN=MIDDLE  ALT=\"n+=+%288+%2B-+2%2Asqrt%2810%29%29%2F-12\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply 2 and -6 to get -12\r<BR>\n" );
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document.write( "So now the expression breaks down into two parts\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+%288+%2B+2%2Asqrt%2810%29%29%2F-12 ALIGN=MIDDLE  ALT=\"n+=+%288+%2B+2%2Asqrt%2810%29%29%2F-12\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+%288+-+2%2Asqrt%2810%29%29%2F-12 ALIGN=MIDDLE  ALT=\"n+=+%288+-+2%2Asqrt%2810%29%29%2F-12\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now break up the fraction\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=%2B8%2F-12%2B2%2Asqrt%2810%29%2F-12 ALIGN=MIDDLE  ALT=\"n=%2B8%2F-12%2B2%2Asqrt%2810%29%2F-12\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=%2B8%2F-12-2%2Asqrt%2810%29%2F-12 ALIGN=MIDDLE  ALT=\"n=%2B8%2F-12-2%2Asqrt%2810%29%2F-12\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Simplify\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=-2+%2F+3-sqrt%2810%29%2F6 ALIGN=MIDDLE  ALT=\"n=-2+%2F+3-sqrt%2810%29%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=-2+%2F+3%2Bsqrt%2810%29%2F6 ALIGN=MIDDLE  ALT=\"n=-2+%2F+3%2Bsqrt%2810%29%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So these expressions approximate to\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=-1.1937129433614 ALIGN=MIDDLE  ALT=\"n=-1.1937129433614\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=-0.139620389971937 ALIGN=MIDDLE  ALT=\"n=-0.139620389971937\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So our solutions are:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=-1.1937129433614 ALIGN=MIDDLE  ALT=\"n=-1.1937129433614\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=-0.139620389971937 ALIGN=MIDDLE  ALT=\"n=-0.139620389971937\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Notice when we graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6%2Ax%5E2-8%2Ax-1 ALIGN=MIDDLE  ALT=\"-6%2Ax%5E2-8%2Ax-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > (just replace n with x), we get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+500%2C+500%2C+-11.1937129433614%2C+9.86037961002806%2C+-11.1937129433614%2C+9.86037961002806%2C-6%2Ax%5E2%2B-8%2Ax%2B-1%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "when we use the root finder feature on a calculator, we find that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1.1937129433614 ALIGN=MIDDLE  ALT=\"x=-1.1937129433614\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-0.139620389971937 ALIGN=MIDDLE  ALT=\"x=-0.139620389971937\"   DISABLED_style= \"max-width:90%; height:auto;\" >.So this verifies our answer </SPAN>\n" );
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