document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=71941>Question 71941</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I am having a real hard time solving this. I can't get the light bulb in my head to go ding when it comes to polynomials etc..Any help will be appreciated.\r<BR>\n" );
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document.write( "6 x ^ 2 - 2 3 x + 1 5 = 0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #51439 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=51439\">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=6x%5E2+-+23x+%2B+15+=+0 ALIGN=MIDDLE  ALT=\"6x%5E2+-+23x+%2B+15+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This can be solved in several ways.  The easiest on ... once you see it is by factoring.<BR>\n" );
document.write( "The left side of this equation factors to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-3%29%2A%286x+-+5%29+=+0 ALIGN=MIDDLE  ALT=\"%28x-3%29%2A%286x+-+5%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This equation will hold true if either of the factors equals zero.  So we can find the answers<BR>\n" );
document.write( "for x by first setting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-3%29+=+0 ALIGN=MIDDLE  ALT=\"%28x-3%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >. By adding 3 to both sides we get as one answer<BR>\n" );
document.write( "to this problem <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+3 ALIGN=MIDDLE  ALT=\"x+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Next set the second factor equal to zero to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x+-+5+=+0 ALIGN=MIDDLE  ALT=\"6x+-+5+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Solve by first adding 5 to both sides to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x+=+5 ALIGN=MIDDLE  ALT=\"6x+=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Then divide both sides by 6 and <BR>\n" );
document.write( "you find the second value of x is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+5%2F6 ALIGN=MIDDLE  ALT=\"x+=+5%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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document.write( "Another way to do this is to use the quadratic formula.  This formula says that if you are<BR>\n" );
document.write( "given a quadratic equation in the general form:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2+%2B+bx+%2B+c+=+0 ALIGN=MIDDLE  ALT=\"ax%5E2+%2B+bx+%2B+c+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "The values of x that satisfy this equation are:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-b%2F%282%2Aa%29+%2B-+sqrt%28b%5E2+-+%284%2Aa%2Ac%29%29%2F%282%2Aa%29 ALIGN=MIDDLE  ALT=\"x+=+-b%2F%282%2Aa%29+%2B-+sqrt%28b%5E2+-+%284%2Aa%2Ac%29%29%2F%282%2Aa%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "By comparing the general form to the problem you were given you can see that a = 6, b = -23, <BR>\n" );
document.write( "and c = 15. Now just plug these values into the equation for x to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-%28-23%29%2F%282%2A6%29+%2B-+sqrt%2823%5E2+-+%284%2A6%2A15%29%29%2F%282%2A6%29 ALIGN=MIDDLE  ALT=\"x+=+-%28-23%29%2F%282%2A6%29+%2B-+sqrt%2823%5E2+-+%284%2A6%2A15%29%29%2F%282%2A6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "First simplifications lead to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+23%2F12+%2B-+sqrt%28529+-+%28360%29%29%2F12 ALIGN=MIDDLE  ALT=\"x+=+23%2F12+%2B-+sqrt%28529+-+%28360%29%29%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "And further:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+23%2F12+%2B-+sqrt%28169%29%2F12 ALIGN=MIDDLE  ALT=\"x+=+23%2F12+%2B-+sqrt%28169%29%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and sqrt(169) = 13 so the problem becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+23%2F12+%2B-+13%2F12 ALIGN=MIDDLE  ALT=\"x+=+23%2F12+%2B-+13%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "So the two answers are:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2823%2B13%29%2F12=+36%2F12+=+3 ALIGN=MIDDLE  ALT=\"x+=+%2823%2B13%29%2F12=+36%2F12+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and <BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2823-13%29%2F12=+10%2F12+=+5%2F6 ALIGN=MIDDLE  ALT=\"x+=+%2823-13%29%2F12=+10%2F12+=+5%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "These answers are the same as we got by factoring.  So they serve to verify that they are<BR>\n" );
document.write( "correct.<BR>\n" );
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document.write( "I hope these couple of ways of doing the problem help you to see how you can solve<BR>\n" );
document.write( "quadratic equations.  Using the quadratic formula is the most general way.  It always works.<BR>\n" );
document.write( "But not all quadratic equations factor nicely. </SPAN>\n" );
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