document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=206019>Question 206019</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0534463428/algebrahomeworkh>Intermediate Algebra</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> x^2 - 3x^1/2 - 10 = 0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #156131 by </B><A HREF=/tutors/aboutme.mpl?userid=mickclns><B>mickclns(59)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=mickclns><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=156131\">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=+x%5E2+-+%283x%5E1%29+%2F+2+-+10+=+0+ ALIGN=MIDDLE  ALT=\"+x%5E2+-+%283x%5E1%29+%2F+2+-+10+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" >  notation housekeeping<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5E2+-+%283%2F2%29x+-+10+=+0+ ALIGN=MIDDLE  ALT=\"+x%5E2+-+%283%2F2%29x+-+10+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" > multiply both sides by 2 to get rid of the fraction<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+2x%5E2+-+3x+-+20+=+0+ ALIGN=MIDDLE  ALT=\"+2x%5E2+-+3x+-+20+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "There are various ways of factoring a quadratic (if it factors, which this does, because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4%2Aa%2Ac+=+9+%2B+160+=+169+ ALIGN=MIDDLE  ALT=\"b%5E2-4%2Aa%2Ac+=+9+%2B+160+=+169+\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a perfect square).  I like this way: <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+2x%5E2+-+3x+-+20+=+0+ ALIGN=MIDDLE  ALT=\"+2x%5E2+-+3x+-+20+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" > First multiply the coefficient of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5E2+ ALIGN=MIDDLE  ALT=\"+x%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > term, 2,  by the constant term, -20, to get -40.  Now you want to find two numbers that multiply to this number, -40, and that add to the coefficient of x, in this case -3. The two numbers are -8 and 5.<BR>\n" );
document.write( "Now you put those two numbers into the blanks in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+2x%5E2+%2B+_x++%2B+_x+-+20+ ALIGN=MIDDLE  ALT=\"+2x%5E2+%2B+_x++%2B+_x+-+20+\"   DISABLED_style= \"max-width:90%; height:auto;\" > (and it doesn't matter which number in which blank):<BR>\n" );
document.write( " <BR>\n" );
document.write( "Case 1: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+2x%5E2+%2B+%28-8%29x++%2B+%285%29x+-+20+ ALIGN=MIDDLE  ALT=\"+2x%5E2+%2B+%28-8%29x++%2B+%285%29x+-+20+\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Now factor out whatever you can out of the first two terms and factor out whatever you can out of the last two terms:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%28x+-+4%29+%2B+5%28x+-+4%29 ALIGN=MIDDLE  ALT=\"2x%28x+-+4%29+%2B+5%28x+-+4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Notice both terms have a factor of x-4 which you can factor out:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x+%2B+5%29+%28x-4%29 ALIGN=MIDDLE  ALT=\"%282x+%2B+5%29+%28x-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  <BR>\n" );
document.write( " <BR>\n" );
document.write( "Case 2: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+2x%5E2+%2B+%285%29x++%2B+%28-8%29x+-+20+ ALIGN=MIDDLE  ALT=\"+2x%5E2+%2B+%285%29x++%2B+%28-8%29x+-+20+\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Now factor out whatever you can out of the first two terms and factor out whatever you can out of the last two terms:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%282x+%2B5%29+%2B+-4%282x+%2B+5%29 ALIGN=MIDDLE  ALT=\"x%282x+%2B5%29+%2B+-4%282x+%2B+5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Again, both terms have a factor, 2x + 5 this time, which you can factor out:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-4%29+%282x+%2B+5%29 ALIGN=MIDDLE  ALT=\"%28x-4%29+%282x+%2B+5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. <BR>\n" );
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document.write( "In both cases the factors are the same except in opposite order, which makes not a whit of difference.<BR>\n" );
document.write( "It is a good idea to multiply out (FOIL) to make sure you have factored correctly<BR>\n" );
document.write( " <BR>\n" );
document.write( "Now we use the zero product property which says that if a product of a bunch of numbers is 0 then one of the numbers must be 0, which is what allows us to set each of the factors to 0.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-4%29+%282x+%2B+5%29+=+0 ALIGN=MIDDLE  ALT=\"%28x-4%29+%282x+%2B+5%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-4+=+0 ALIGN=MIDDLE  ALT=\"x-4+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2B5+=+0 ALIGN=MIDDLE  ALT=\"2x%2B5+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+4 ALIGN=MIDDLE  ALT=\"x+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-5%2F2 ALIGN=MIDDLE  ALT=\"x+=+-5%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  \r<BR>\n" );
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document.write( "Now it's a good idea to check your answers to make sure they work, by plugging them in (one at a time) for x in the ORIGINAL equation (I will do one, you can do the other):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28-5%2F2%29%5E2+-+%283%28-5%2F2%29%5E1%29+%2F+2+-+10+=+0+ ALIGN=MIDDLE  ALT=\"+%28-5%2F2%29%5E2+-+%283%28-5%2F2%29%5E1%29+%2F+2+-+10+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" > ?<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+25%2F4+%2B+15%2F4+-+10+=+0 ALIGN=MIDDLE  ALT=\"+25%2F4+%2B+15%2F4+-+10+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >?<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=40%2F4+-+10+=+0 ALIGN=MIDDLE  ALT=\"40%2F4+-+10+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >?<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=+0 ALIGN=MIDDLE  ALT=\"0+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Yes! </SPAN>\n" );
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