document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=695874>Question 695874</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me solve by completing the square:\r<BR>\n" );
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document.write( "x^2 + 15 x = -56 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #428665 by </B><A HREF=/tutors/aboutme.mpl?userid=mouk><B>mouk(232)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=mouk><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=mouk><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=428665\">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=+x%5E2+%2B+15x+=+-56+ ALIGN=MIDDLE  ALT=\"+x%5E2+%2B+15x+=+-56+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5E2+%2B+15x+%2B+56+=+0+ ALIGN=MIDDLE  ALT=\"+x%5E2+%2B+15x+%2B+56+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\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=1z%5E2%2B15z%2B56=0 ALIGN=MIDDLE  ALT=\"1z%5E2%2B15z%2B56=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=1z%5E2%2B15z%2B56=0 ALIGN=MIDDLE  ALT=\"1z%5E2%2B15z%2B56=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=%28z%2Bsomenumber%29%5E2+%2B+othernumber ALIGN=MIDDLE  ALT=\"%28z%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=%28z%2Bsomenumber%29%5E2 ALIGN=MIDDLE  ALT=\"%28z%2Bsomenumber%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28z%2Bsomenumber%29%5E2+=+z%5E2%2B2%2Asomenumber%2Ax+%2B+somenumber%5E2 ALIGN=MIDDLE  ALT=\"%28z%2Bsomenumber%29%5E2+=+z%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=1z%5E2%2Bhighlight_red%28+15%29+%2A+z%2B56=0 ALIGN=MIDDLE  ALT=\"1z%5E2%2Bhighlight_red%28+15%29+%2A+z%2B56=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > that goes in front of z is 15, we know that 15=2*somenumber, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=somenumber+=+15%2F2 ALIGN=MIDDLE  ALT=\"somenumber+=+15%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=%28z%2B15%2F2%29%5E2+%2B+othernumber ALIGN=MIDDLE  ALT=\"%28z%2B15%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=%28z%2B15%2F2%29%5E2+%2B+othernumber ALIGN=MIDDLE  ALT=\"%28z%2B15%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=1z%5E2%2B15z%2Bhighlight_green%28+56+%29=0 ALIGN=MIDDLE  ALT=\"1z%5E2%2B15z%2Bhighlight_green%28+56+%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=%28z%2B15%2F2%29%5E2+%2B+othernumber+=+z%5E2%2B15z%2Bhighlight_red%28+15%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 56 (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=15%5E2%2F4+%2B+othernumber+=+56 ALIGN=MIDDLE  ALT=\"15%5E2%2F4+%2B+othernumber+=+56\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=othernumber+=+56-15%5E2%2F4+=+-0.25 ALIGN=MIDDLE  ALT=\"othernumber+=+56-15%5E2%2F4+=+-0.25\"   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=%28z%2B15%2F2%29%5E2+%2B+-0.25+=+0 ALIGN=MIDDLE  ALT=\"%28z%2B15%2F2%29%5E2+%2B+-0.25+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28z%2B15%2F2%29%5E2+=+0.25 ALIGN=MIDDLE  ALT=\"%28z%2B15%2F2%29%5E2+=+0.25\"   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=%28z%2B15%2F2%29%5E2 ALIGN=MIDDLE  ALT=\"%28z%2B15%2F2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, equated to a number (0.25).<BR>\n" );
document.write( "  <BR>\n" );
document.write( "    Since the right part 0.25 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+%28z%2B15%2F2%29+=+%2Bsqrt%28+0.25+%29%2C+%28z%2B15%2F2%29+=+-sqrt%28+0.25+%29+%29 ALIGN=MIDDLE  ALT=\"system%28+%28z%2B15%2F2%29+=+%2Bsqrt%28+0.25+%29%2C+%28z%2B15%2F2%29+=+-sqrt%28+0.25+%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+%28z%2B15%2F2%29+=+0.5%2C+%28z%2B15%2F2%29+=+-0.5+%29 ALIGN=MIDDLE  ALT=\"system%28+%28z%2B15%2F2%29+=+0.5%2C+%28z%2B15%2F2%29+=+-0.5+%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+z%2B15%2F2+=+0.5%2C+z%2B15%2F2+=+-0.5+%29 ALIGN=MIDDLE  ALT=\"system%28+z%2B15%2F2+=+0.5%2C+z%2B15%2F2+=+-0.5+%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+z+=+0.5-15%2F2%2C+z+=+-0.5-15%2F2+%29 ALIGN=MIDDLE  ALT=\"system%28+z+=+0.5-15%2F2%2C+z+=+-0.5-15%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+z+=+-7%2C+z+=+-8+%29 ALIGN=MIDDLE  ALT=\"system%28+z+=+-7%2C+z+=+-8+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Answer: z=-7, -8.\n" );
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