document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=4794>Question 4794</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A quadratic equation always has two solutions. How do you account for<BR>\n" );
document.write( "this difference when compared to linear equations? At what step in the<BR>\n" );
document.write( "process do we ensure that we get two solutions when using the<BR>\n" );
document.write( "Completing the Square method?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #2295 by </B><A HREF=/tutors/aboutme.mpl?userid=rapaljer><B>rapaljer(4671)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=rapaljer><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=rapaljer><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=2295\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> In any polynomial equation, the degree or order of the polynomial is the highest power of the variable in the equation.  The degree of the polynomial gives you the total number of solutions to the equation.  For example, a linear equation is a simple \"x-to-the-first-power\" equation, has one solution.  A quadratic equation is of second degree, so it has two solutions.  A cubic equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3 ALIGN=MIDDLE  ALT=\"x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > will have three solutions, etc.\r<BR>\n" );
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document.write( "However, having said that, it is possible that a quadratic equation might have two solutions that are actually the same number.  For example, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+6x+%2B+9=0 ALIGN=MIDDLE  ALT=\"x%5E2+-+6x+%2B+9=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be factored into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-3%29%28x-3%29+=+0 ALIGN=MIDDLE  ALT=\"x-3%29%28x-3%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28x-3%29%5E2+=+0 ALIGN=MIDDLE  ALT=\"+%28x-3%29%5E2+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  In this case, the TWO solutions are x= 3 and x=3.  We say that this is a \"double root (solution)\" or that the solution x=3 is of \"multiplicity 2.\"   Keep in mind that the two solutions in a quadratic equation might be complex solutions, and in this case there are no real solutions at all.\r<BR>\n" );
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document.write( "In completing the square process, when you have something like <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28x-5%29%5E2+=+7 ALIGN=MIDDLE  ALT=\"+%28x-5%29%5E2+=+7\"   DISABLED_style= \"max-width:90%; height:auto;\" > and you must take the square root of both sides to \"undo\" the square in order to solve for x, you MUST put the PLUS or MINUS in order to get both solutions.  The answer here will be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x-5+=+0+%2B-+sqrt+%287%29 ALIGN=MIDDLE  ALT=\"+x-5+=+0+%2B-+sqrt+%287%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Finally add 5 to each side of the equation to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+5+%2B-+sqrt+%287%29+ ALIGN=MIDDLE  ALT=\"x+=+5+%2B-+sqrt+%287%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "which is actually TWO solutions.    If you forget the PLUS or MINUS, then you automatically lose half of your answers.\r<BR>\n" );
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document.write( "R^2 at SCC </SPAN>\n" );
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