document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=131609>Question 131609</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I have to factor this problem and I have not found a solution for it. <BR>\n" );
document.write( "8ksquared-19k+9 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #96088 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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=96088\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Given to factor:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8k%5E2+-+19k+%2B+9 ALIGN=MIDDLE  ALT=\"8k%5E2+-+19k+%2B+9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This does not factor very \"nicely\". The way to factor this is to use the quadratic equation.<BR>\n" );
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document.write( "I'm going to presume that you know how to use the quadratic equation. When you have a quadratic<BR>\n" );
document.write( "of the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8k%5E2+-+19k+%2B+9+=+0 ALIGN=MIDDLE  ALT=\"8k%5E2+-+19k+%2B+9+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > you can find the solutions for k by using the quadratic formula:<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( "where a is the multiplier of the squared term (+8 in this problem), b is the multiplier of<BR>\n" );
document.write( "the first degree term (-19 in this problem), and c is the constant term (+9 in this problem).<BR>\n" );
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document.write( "If you substitute these values into the quadratic formula and simplify the resulting<BR>\n" );
document.write( "equation for k you will find that k has two answers:<BR>\n" );
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document.write( "k = 1.7215002 and k = 0.65349977<BR>\n" );
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document.write( "This means that factors of the given problem include the two factors:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28k+-+1.7215002%29 ALIGN=MIDDLE  ALT=\"%28k+-+1.7215002%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28k+-+0.65349977%29 ALIGN=MIDDLE  ALT=\"%28k+-+0.65349977%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "But if you multiply these two factors together you get back to the equation:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+2.3749999x+%2B+1.124999985+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+-+2.3749999x+%2B+1.124999985+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Notice that the left side ... which is equal to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28k-1.7215002%29%2A%28k-0.65349977%29 ALIGN=MIDDLE  ALT=\"%28k-1.7215002%29%2A%28k-0.65349977%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > must be multiplied <BR>\n" );
document.write( "by 8 to equal the original equation that we started with. Therefore, our equation is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%2A%28k-1.7215002%29%2A%28k-0.65349977%29+=+0 ALIGN=MIDDLE  ALT=\"8%2A%28k-1.7215002%29%2A%28k-0.65349977%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "So our factors of the original problem that you were given are:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%2A%28k-1.7215002%29%2A%28k-0.65349977%29 ALIGN=MIDDLE  ALT=\"8%2A%28k-1.7215002%29%2A%28k-0.65349977%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "And if you multiply these all together you will get very, very close to the original expression.<BR>\n" );
document.write( "You will be within the round-off error of the two decimal numbers.<BR>\n" );
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document.write( "Hope this doesn't confuse you too much. If a quadratic expression is too difficult to<BR>\n" );
document.write( "factor you can always use this general method. Use the quadratic formula to find the roots, then <BR>\n" );
document.write( "convert the roots to the factor form, and finally multiply by an appropriate constant to make<BR>\n" );
document.write( "the two factors work out to the original expression.<BR>\n" );
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document.write( "Good luck with studying this to see how it works.<BR>\n" );
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