document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=119097>Question 119097</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Determine the value of 'k' if the product of the roots of 3kx^2+2(x-1)+k=0 is 1. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #87259 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=87259\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3kx%5E2%2B2%28x-1%29%2Bk=0 ALIGN=MIDDLE  ALT=\"3kx%5E2%2B2%28x-1%29%2Bk=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Step 1:  Put the equation into standard form, i.e. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=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=3kx%5E2%2B2x-2%2Bk=0 ALIGN=MIDDLE  ALT=\"3kx%5E2%2B2x-2%2Bk=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or more neatly:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3kx%5E2%2B2x%2Bk-2=0 ALIGN=MIDDLE  ALT=\"3kx%5E2%2B2x%2Bk-2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Step 2:  Apply the quadratic formula with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=3k ALIGN=MIDDLE  ALT=\"a=3k\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=2 ALIGN=MIDDLE  ALT=\"b=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=k-2 ALIGN=MIDDLE  ALT=\"c=k-2\"   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=x+=+%28-2+%2B-+sqrt%28+4-4%2A%283k%29%2A%28k-2%29+%29%29%2F%282%2A3k%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-2+%2B-+sqrt%28+4-4%2A%283k%29%2A%28k-2%29+%29%29%2F%282%2A3k%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "Step 3: Simplify\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-2+%2B-+sqrt%28+4-12k%5E2%2B24k%29+%29%2F%286k%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-2+%2B-+sqrt%28+4-12k%5E2%2B24k%29+%29%2F%286k%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Step 4: We are looking for k such that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5Dx%5B2%5D=1 ALIGN=MIDDLE  ALT=\"x%5B1%5Dx%5B2%5D=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28-2+%2B+sqrt%28+4-12k%5E2%2B24k%29+%29%2F%286k%29%29%28%28-2+-+sqrt%28+4-12k%5E2%2B24k%29+%29%2F%286k%29%29+=1 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Step 5:  This looks like a hideous mess to multiply, but remember that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2Bb%29%28a-b%29=a%5E2-b%5E2 ALIGN=MIDDLE  ALT=\"%28a%2Bb%29%28a-b%29=a%5E2-b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284-%284-12k%5E2%2B24k%29%29%2F36k%5E2=1 ALIGN=MIDDLE  ALT=\"%284-%284-12k%5E2%2B24k%29%29%2F36k%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Step 6: Simplify and solve\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2812k%5E2-24k%29%2F36k%5E2=1 ALIGN=MIDDLE  ALT=\"%2812k%5E2-24k%29%2F36k%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12k%5E2-24k=36k%5E2 ALIGN=MIDDLE  ALT=\"12k%5E2-24k=36k%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-24k%5E2-24k=0 ALIGN=MIDDLE  ALT=\"-24k%5E2-24k=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k%5E2%2Bk=0 ALIGN=MIDDLE  ALT=\"k%5E2%2Bk=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k%28k%2B1%29=0 ALIGN=MIDDLE  ALT=\"k%28k%2B1%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+0 ALIGN=MIDDLE  ALT=\"k+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-1 ALIGN=MIDDLE  ALT=\"k=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Step 7:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=0 ALIGN=MIDDLE  ALT=\"k=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be excluded because that would make the lead coefficient on the original quadradic go to zero.  Therefore, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-1 ALIGN=MIDDLE  ALT=\"k=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Step 8:  Check the answer.  Using <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-1 ALIGN=MIDDLE  ALT=\"k=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the original quadratic becomes\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3x%5E2%2B2x-3=0 ALIGN=MIDDLE  ALT=\"-3x%5E2%2B2x-3=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=x+=+%28-2+%2B-+sqrt%28+4-4%28-3%29%28-3%29%29%29%2F%28-6%29=%28-2%2B-sqrt%28-32%29%29%2F-6=%28-2%2B-sqrt%282%294i%29%2F6+=%28-1%2B-sqrt%282%292i%29%2F3 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The product of these two roots should be 1:  Does <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28-1%2Bsqrt%282%292i%29%2F3%29%28%28-1-sqrt%282%292i%29%2F3%29=1 ALIGN=MIDDLE  ALT=\"%28%28-1%2Bsqrt%282%292i%29%2F3%29%28%28-1-sqrt%282%292i%29%2F3%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >?\r<BR>\n" );
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document.write( "Again, we can simplify something that looks rather messy with the difference of two squares factorization, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2Bb%29%28a-b%29=a%5E2-b%5E2 ALIGN=MIDDLE  ALT=\"%28a%2Bb%29%28a-b%29=a%5E2-b%5E2\"   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=%28%28-1%2Bsqrt%282%292i%29%2F3%29%28%28-1-sqrt%282%292i%29%2F3%29=%281-%282%29%284%29%28-1%29%29%2F9=9%2F9=1 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >:  Answer checks. </SPAN>\n" );
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