document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=122284>Question 122284</A>: <I><SPAN STYLE=\"word-break: break-all;\"> x4 – 3 x2 + 2 = 0.<BR>\n" );
document.write( "Explain how one can use the quadratic formula to solve this equation by using a change of variable. Solve this equation completely.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #89773 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=89773\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u=x%5E2 ALIGN=MIDDLE  ALT=\"u=x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get\r<BR>\n" );
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document.write( "Let's use the quadratic formula to solve for u:\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( "So lets solve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u%5E2-3%2Au%2B2=0 ALIGN=MIDDLE  ALT=\"u%5E2-3%2Au%2B2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ( notice <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=1 ALIGN=MIDDLE  ALT=\"a=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-3 ALIGN=MIDDLE  ALT=\"b=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=2 ALIGN=MIDDLE  ALT=\"c=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=u+=+%28--3+%2B-+sqrt%28+%28-3%29%5E2-4%2A1%2A2+%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"u+=+%28--3+%2B-+sqrt%28+%28-3%29%5E2-4%2A1%2A2+%29%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in a=1, b=-3, and c=2\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u+=+%283+%2B-+sqrt%28+%28-3%29%5E2-4%2A1%2A2+%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"u+=+%283+%2B-+sqrt%28+%28-3%29%5E2-4%2A1%2A2+%29%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Negate -3 to get 3\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u+=+%283+%2B-+sqrt%28+9-4%2A1%2A2+%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"u+=+%283+%2B-+sqrt%28+9-4%2A1%2A2+%29%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square -3 to get 9  (note: remember when you square -3, you must square the negative as well. This is because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-3%29%5E2=-3%2A-3=9 ALIGN=MIDDLE  ALT=\"%28-3%29%5E2=-3%2A-3=9\"   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=u+=+%283+%2B-+sqrt%28+9%2B-8+%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"u+=+%283+%2B-+sqrt%28+9%2B-8+%29%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2A2%2A1 ALIGN=MIDDLE  ALT=\"-4%2A2%2A1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-8 ALIGN=MIDDLE  ALT=\"-8\"   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=u+=+%283+%2B-+sqrt%28+1+%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"u+=+%283+%2B-+sqrt%28+1+%29%29%2F%282%2A1%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=u+=+%283+%2B-+1%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"u+=+%283+%2B-+1%29%2F%282%2A1%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=u+=+%283+%2B-+1%29%2F2 ALIGN=MIDDLE  ALT=\"u+=+%283+%2B-+1%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply 2 and 1 to get 2\r<BR>\n" );
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document.write( "So now the expression breaks down into two parts\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u+=+%283+%2B+1%29%2F2 ALIGN=MIDDLE  ALT=\"u+=+%283+%2B+1%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u+=+%283+-+1%29%2F2 ALIGN=MIDDLE  ALT=\"u+=+%283+-+1%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Lets look at the first part:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%283+%2B+1%29%2F2 ALIGN=MIDDLE  ALT=\"x=%283+%2B+1%29%2F2\"   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=u=4%2F2 ALIGN=MIDDLE  ALT=\"u=4%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add the terms in the numerator\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u=2 ALIGN=MIDDLE  ALT=\"u=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide\r<BR>\n" );
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document.write( "So one answer is\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u=2 ALIGN=MIDDLE  ALT=\"u=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now lets look at the second part:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%283+-+1%29%2F2 ALIGN=MIDDLE  ALT=\"x=%283+-+1%29%2F2\"   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=u=2%2F2 ALIGN=MIDDLE  ALT=\"u=2%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract the terms in the numerator\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u=1 ALIGN=MIDDLE  ALT=\"u=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide\r<BR>\n" );
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document.write( "So another answer is\r<BR>\n" );
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document.write( "So our solutions are:\r<BR>\n" );
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document.write( "Now remember, we let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u=x%5E2 ALIGN=MIDDLE  ALT=\"u=x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So our solutions become\r<BR>\n" );
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document.write( "Now take the square root of both sides for each case\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So our answers are\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=sqrt%282%29 ALIGN=MIDDLE  ALT=\"x=sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-sqrt%282%29 ALIGN=MIDDLE  ALT=\"x=-sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1 ALIGN=MIDDLE  ALT=\"x=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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