document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=93575>Question 93575</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use the quadratic formula to solve the following equation.\r<BR>\n" );
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document.write( "3x^2-2x+1=0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #68147 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=68147\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's use the quadratic formula to solve for x:\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=x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29 ALIGN=MIDDLE  ALT=\"x+=+%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=3%2Ax%5E2-2%2Ax%2B1=0 ALIGN=MIDDLE  ALT=\"3%2Ax%5E2-2%2Ax%2B1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ( notice <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=3 ALIGN=MIDDLE  ALT=\"a=3\"   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=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=x+=+%28--2+%2B-+sqrt%28+%28-2%29%5E2-4%2A3%2A1+%29%29%2F%282%2A3%29 ALIGN=MIDDLE  ALT=\"x+=+%28--2+%2B-+sqrt%28+%28-2%29%5E2-4%2A3%2A1+%29%29%2F%282%2A3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in a=3, b=-2, and c=1\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%282+%2B-+sqrt%28+%28-2%29%5E2-4%2A3%2A1+%29%29%2F%282%2A3%29 ALIGN=MIDDLE  ALT=\"x+=+%282+%2B-+sqrt%28+%28-2%29%5E2-4%2A3%2A1+%29%29%2F%282%2A3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Negate -2 to get 2\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%282+%2B-+sqrt%28+4-4%2A3%2A1+%29%29%2F%282%2A3%29 ALIGN=MIDDLE  ALT=\"x+=+%282+%2B-+sqrt%28+4-4%2A3%2A1+%29%29%2F%282%2A3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square -2 to get 4  (note: remember when you square -2, you must square the negative as well. This is because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-2%29%5E2=-2%2A-2=4 ALIGN=MIDDLE  ALT=\"%28-2%29%5E2=-2%2A-2=4\"   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+=+%282+%2B-+sqrt%28+4%2B-12+%29%29%2F%282%2A3%29 ALIGN=MIDDLE  ALT=\"x+=+%282+%2B-+sqrt%28+4%2B-12+%29%29%2F%282%2A3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2A1%2A3 ALIGN=MIDDLE  ALT=\"-4%2A1%2A3\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-12 ALIGN=MIDDLE  ALT=\"-12\"   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+=+%282+%2B-+sqrt%28+-8+%29%29%2F%282%2A3%29 ALIGN=MIDDLE  ALT=\"x+=+%282+%2B-+sqrt%28+-8+%29%29%2F%282%2A3%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=x+=+%282+%2B-+2%2Ai%2Asqrt%282%29%29%2F%282%2A3%29 ALIGN=MIDDLE  ALT=\"x+=+%282+%2B-+2%2Ai%2Asqrt%282%29%29%2F%282%2A3%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=x+=+%282+%2B-+2%2Ai%2Asqrt%282%29%29%2F%286%29 ALIGN=MIDDLE  ALT=\"x+=+%282+%2B-+2%2Ai%2Asqrt%282%29%29%2F%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply 2 and 3 to get 6\r<BR>\n" );
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document.write( "After simplifying, the quadratic has roots of\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1%2F3+%2B+sqrt%282%29%2F3%2Ai ALIGN=MIDDLE  ALT=\"x=1%2F3+%2B+sqrt%282%29%2F3%2Ai\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1%2F3+-+sqrt%282%29%2F3%2Ai ALIGN=MIDDLE  ALT=\"x=1%2F3+-+sqrt%282%29%2F3%2Ai\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Notice if we graph the quadratic <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=3%2Ax%5E2-2%2Ax%2B1 ALIGN=MIDDLE  ALT=\"y=3%2Ax%5E2-2%2Ax%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, 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+-14.6666666666667%2C+15.3333333333333%2C+-14.3333333333333%2C+15.6666666666667%2C+3%2Ax%5E2-2%2Ax%2B1%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=3%2Ax%5E2-2%2Ax%2B1 ALIGN=MIDDLE  ALT=\"y=3%2Ax%5E2-2%2Ax%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "And we can see that there are no real roots\r<BR>\n" );
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document.write( "To visually verify the answer, check out <a  rel=nofollow HREF=http://www.math.hmc.edu/funfacts/ffiles/10005.1.shtml>this page</a> to see a visual representation of imaginary roots<BR>\n" );
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