document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=980552>Question 980552</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve this equation by completing the square. <BR>\n" );
document.write( "3x^2 - 4x = -2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #601670 by </B><A HREF=/tutors/aboutme.mpl?userid=richwmiller><B>richwmiller(17219)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richwmiller><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/cgi-bin/show-face.mpl?user=richwmiller><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=601670\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <TABLE  BORDER=1>\n" );
document.write( "\n" );
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=3x%5E2%2B-4x%2B2=0 ALIGN=MIDDLE  ALT=\"3x%5E2%2B-4x%2B2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > to standard form by dividing both sides by 3: <BR>\n" );
document.write( "We have: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%5E2%2B-1.33333333333333x%2B0.666666666666667=0 ALIGN=MIDDLE  ALT=\"1x%5E2%2B-1.33333333333333x%2B0.666666666666667=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+-1.33333333333333%29+%2A+x%2B0.666666666666667=0 ALIGN=MIDDLE  ALT=\"1x%5E2%2Bhighlight_red%28+-1.33333333333333%29+%2A+x%2B0.666666666666667=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > that goes in front of x is -1.33333333333333, we know that -1.33333333333333=2*somenumber, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=somenumber+=+-1.33333333333333%2F2 ALIGN=MIDDLE  ALT=\"somenumber+=+-1.33333333333333%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-1.33333333333333%2F2%29%5E2+%2B+othernumber ALIGN=MIDDLE  ALT=\"%28x%2B-1.33333333333333%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-1.33333333333333%2F2%29%5E2+%2B+othernumber ALIGN=MIDDLE  ALT=\"%28x%2B-1.33333333333333%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-1.33333333333333x%2Bhighlight_green%28+0.666666666666667+%29=0 ALIGN=MIDDLE  ALT=\"1x%5E2%2B-1.33333333333333x%2Bhighlight_green%28+0.666666666666667+%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-1.33333333333333%2F2%29%5E2+%2B+othernumber+=+x%5E2%2B-1.33333333333333x%2Bhighlight_red%28+-1.33333333333333%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 0.666666666666667 (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=-1.33333333333333%5E2%2F4+%2B+othernumber+=+0.666666666666667 ALIGN=MIDDLE  ALT=\"-1.33333333333333%5E2%2F4+%2B+othernumber+=+0.666666666666667\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=othernumber+=+0.666666666666667--1.33333333333333%5E2%2F4+=+0.222222222222222 ALIGN=MIDDLE  ALT=\"othernumber+=+0.666666666666667--1.33333333333333%5E2%2F4+=+0.222222222222222\"   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-1.33333333333333%2F2%29%5E2+%2B+0.222222222222222+=+0 ALIGN=MIDDLE  ALT=\"%28x%2B-1.33333333333333%2F2%29%5E2+%2B+0.222222222222222+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B-1.33333333333333%2F2%29%5E2+=+-0.222222222222222 ALIGN=MIDDLE  ALT=\"%28x%2B-1.33333333333333%2F2%29%5E2+=+-0.222222222222222\"   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-1.33333333333333%2F2%29%5E2 ALIGN=MIDDLE  ALT=\"%28x%2B-1.33333333333333%2F2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, equated to a number (-0.222222222222222).<BR>\n" );
document.write( "  <BR>\n" );
document.write( "    There is no number whose square can be negative. So, there is no solution to this equation</TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "\n" );
document.write( "<TABLE  BORDER=1>\n" );
document.write( "\n" );
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=3x%5E2%2B-4x%2B2+=+0 ALIGN=MIDDLE  ALT=\"3x%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%2A3%2A2=-8 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-4%29%5E2-4%2A3%2A2=-8\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  The discriminant -8 is less than zero. That means that there are no solutions among real numbers.<BR><BR>\n" );
document.write( "  If you are a student of advanced school algebra and are aware about <u>imaginary numbers<u>, read on.<BR><BR>\n" );
document.write( "  <BR>\n" );
document.write( "      In the field of imaginary numbers, the square root of -8 is + or - <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28+8%29+=+2.82842712474619 ALIGN=MIDDLE  ALT=\"sqrt%28+8%29+=+2.82842712474619\"   DISABLED_style= \"max-width:90%; height:auto;\" >. <BR>\n" );
document.write( "  <BR>\n" );
document.write( "      The solution is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B12%5D+=+%28--4%2B-+i%2Asqrt%28+-8+%29%29%2F2%5C3+=++%28--4%2B-+i%2A2.82842712474619%29%2F2%5C3+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\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+3%2Ax%5E2%2B-4%2Ax%2B2+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+3%2Ax%5E2%2B-4%2Ax%2B2+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
document.write( "</TABLE>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );