document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=43566>Question 43566</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use completing the square or the quadradic formula to find the solutions to the equation: x^2-4x+2=0.  Thank you for your time. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #28565 by </B><A HREF=/tutors/aboutme.mpl?userid=aaaaaaaa><B>aaaaaaaa(138)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=aaaaaaaa><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=28565\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <TABLE  BORDER=1>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=quadratic.solver>SOLVE quadratic equation with variable</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Quadratic equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%5E2%2B-4x%2B2+=+0 ALIGN=MIDDLE  ALT=\"1x%5E2%2B-4x%2B2+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) has the following solutons:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  For these solutions to exist, the <U>discriminant</U> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > should not be a negative number.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  First, we need to compute the discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac=%28-4%29%5E2-4%2A1%2A2=8 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-4%29%5E2-4%2A1%2A2=8\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=8 is greater than zero. That means that there are two solutions: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5B12%5D+=+%28--4%2B-sqrt%28+8+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28--4%2B-sqrt%28+8+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D+=+%28-%28-4%29%2Bsqrt%28+8+%29%29%2F2%5C1+=+3.41421356237309 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%28-4%29%2Bsqrt%28+8+%29%29%2F2%5C1+=+3.41421356237309\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D+=+%28-%28-4%29-sqrt%28+8+%29%29%2F2%5C1+=+0.585786437626905 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%28-4%29-sqrt%28+8+%29%29%2F2%5C1+=+0.585786437626905\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <BR>\n" );
document.write( "    Quadratic expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%5E2%2B-4x%2B2 ALIGN=MIDDLE  ALT=\"1x%5E2%2B-4x%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be factored:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%5E2%2B-4x%2B2+=+1%28x-3.41421356237309%29%2A%28x-0.585786437626905%29 ALIGN=MIDDLE  ALT=\"1x%5E2%2B-4x%2B2+=+1%28x-3.41421356237309%29%2A%28x-0.585786437626905%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 3.41421356237309, 0.585786437626905.\n" );
document.write( "Here's your graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-4%2Ax%2B2+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-4%2Ax%2B2+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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document.write( "Now for the other method:\r<BR>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=complete_square.solver>COMPLETING THE SQUARE solver for quadratics</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Read <A  rel=nofollow HREF=Fool-Proof-Completing-the-Square-Part-I.lesson>this lesson on completing the square by prince_abubu</A>, if you do not know how to complete the square.<BR>Let's convert <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%5E2%2B-4x%2B2=0 ALIGN=MIDDLE  ALT=\"1x%5E2%2B-4x%2B2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > to standard form by dividing both sides by 1: <BR>\n" );
document.write( "We have: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%5E2%2B-4x%2B2=0 ALIGN=MIDDLE  ALT=\"1x%5E2%2B-4x%2B2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \n" );
document.write( "What we want to do now is to change this equation to a complete square <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2Bsomenumber%29%5E2+%2B+othernumber ALIGN=MIDDLE  ALT=\"%28x%2Bsomenumber%29%5E2+%2B+othernumber\"   DISABLED_style= \"max-width:90%; height:auto;\" >. How can we find out values of <B>somenumber</B> and <B>othernumber</B> that would make it work?<BR>\n" );
document.write( "Look at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2Bsomenumber%29%5E2 ALIGN=MIDDLE  ALT=\"%28x%2Bsomenumber%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2Bsomenumber%29%5E2+=+x%5E2%2B2%2Asomenumber%2Ax+%2B+somenumber%5E2 ALIGN=MIDDLE  ALT=\"%28x%2Bsomenumber%29%5E2+=+x%5E2%2B2%2Asomenumber%2Ax+%2B+somenumber%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Since the coefficient in our equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%5E2%2Bhighlight_red%28+-4%29+%2A+x%2B2=0 ALIGN=MIDDLE  ALT=\"1x%5E2%2Bhighlight_red%28+-4%29+%2A+x%2B2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > that goes in front of x is -4, we know that -4=2*somenumber, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=somenumber+=+-4%2F2 ALIGN=MIDDLE  ALT=\"somenumber+=+-4%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So, we know that our equation can be rewritten as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B-4%2F2%29%5E2+%2B+othernumber ALIGN=MIDDLE  ALT=\"%28x%2B-4%2F2%29%5E2+%2B+othernumber\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and we do not yet know the other number.<BR>\n" );
document.write( "We are almost there. Finding the other number is simply a matter of not making too many mistakes. We need to find 'other number' such that  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B-4%2F2%29%5E2+%2B+othernumber ALIGN=MIDDLE  ALT=\"%28x%2B-4%2F2%29%5E2+%2B+othernumber\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to our original equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%5E2%2B-4x%2Bhighlight_green%28+2+%29=0 ALIGN=MIDDLE  ALT=\"1x%5E2%2B-4x%2Bhighlight_green%28+2+%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B-4%2F2%29%5E2+%2B+othernumber+=+x%5E2%2B-4x%2Bhighlight_red%28+-4%5E2%2F4%2Bothernumber+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  The highlighted <FONT COLOR=red>red part</FONT> must be equal to 2 (highlighted <FONT COLOR=green>green part</FONT>).<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%5E2%2F4+%2B+othernumber+=+2 ALIGN=MIDDLE  ALT=\"-4%5E2%2F4+%2B+othernumber+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=othernumber+=+2--4%5E2%2F4+=+-2 ALIGN=MIDDLE  ALT=\"othernumber+=+2--4%5E2%2F4+=+-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "So, the equation converts to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B-4%2F2%29%5E2+%2B+-2+=+0 ALIGN=MIDDLE  ALT=\"%28x%2B-4%2F2%29%5E2+%2B+-2+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B-4%2F2%29%5E2+=+2 ALIGN=MIDDLE  ALT=\"%28x%2B-4%2F2%29%5E2+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "    Our equation converted to a square <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B-4%2F2%29%5E2 ALIGN=MIDDLE  ALT=\"%28x%2B-4%2F2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, equated to a number (2).<BR>\n" );
document.write( "  <BR>\n" );
document.write( "    Since the right part 2 is greater than zero, there are two solutions:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "     <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28+%28x%2B-4%2F2%29+=+%2Bsqrt%28+2+%29%2C+%28x%2B-4%2F2%29+=+-sqrt%28+2+%29+%29 ALIGN=MIDDLE  ALT=\"system%28+%28x%2B-4%2F2%29+=+%2Bsqrt%28+2+%29%2C+%28x%2B-4%2F2%29+=+-sqrt%28+2+%29+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    , or<BR>\n" );
document.write( "  <BR>\n" );
document.write( "     <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28+%28x%2B-4%2F2%29+=+1.4142135623731%2C+%28x%2B-4%2F2%29+=+-1.4142135623731+%29 ALIGN=MIDDLE  ALT=\"system%28+%28x%2B-4%2F2%29+=+1.4142135623731%2C+%28x%2B-4%2F2%29+=+-1.4142135623731+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "     <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28+x%2B-4%2F2+=+1.4142135623731%2C+x%2B-4%2F2+=+-1.4142135623731+%29 ALIGN=MIDDLE  ALT=\"system%28+x%2B-4%2F2+=+1.4142135623731%2C+x%2B-4%2F2+=+-1.4142135623731+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "     <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28+x+=+1.4142135623731--4%2F2%2C+x+=+-1.4142135623731--4%2F2+%29 ALIGN=MIDDLE  ALT=\"system%28+x+=+1.4142135623731--4%2F2%2C+x+=+-1.4142135623731--4%2F2+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <BR>\n" );
document.write( "     <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28+x+=+3.41421356237309%2C+x+=+0.585786437626905+%29 ALIGN=MIDDLE  ALT=\"system%28+x+=+3.41421356237309%2C+x+=+0.585786437626905+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Answer: x=3.41421356237309, 0.585786437626905.\n" );
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document.write( "Note that it says that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+2+%2B-+sqrt%282%29 ALIGN=MIDDLE  ALT=\"x+=+2+%2B-+sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  </SPAN>\n" );
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