document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=83586>Question 83586</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Help Please.\r<BR>\n" );
document.write( "\n" );
document.write( "determine whether each of the following trinomials is a perfect square. If it is, factor the trinomials.\r<BR>\n" );
document.write( "\n" );
document.write( "x^2-24x+48 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #60104 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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=60104\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Is:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-24x%2B48 ALIGN=MIDDLE  ALT=\"x%5E2-24x%2B48\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "a perfect square?  The answer is, \"No, it is not.\"<BR>\n" );
document.write( ".<BR>\n" );
document.write( "But how do you easily get that answer.  The method to use is called \"Completing the Square.\"<BR>\n" );
document.write( ".<BR>\n" );
document.write( "How do you use that method?  First, make sure that the coefficient of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > term<BR>\n" );
document.write( "is 1.  [In this case it is, otherwise you would have to factor out from all the terms in<BR>\n" );
document.write( "the trinomial the coefficient (multiplier) of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > term.]<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Next, forget about the +48 for the time being.  Take half of the multiplier of the x term.  That<BR>\n" );
document.write( "multiplier is -24 and half of it is -12.  Then square the -12 to get + 144.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Now we can say that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+24x+%2B+144 ALIGN=MIDDLE  ALT=\"x%5E2+-+24x+%2B+144\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the perfect square trinomial. [Its equivalent<BR>\n" );
document.write( "form is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-12%29%5E2 ALIGN=MIDDLE  ALT=\"%28x-12%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.] So for the given trinomial to be a perfect square, the +48 would<BR>\n" );
document.write( "have to be changed to +144.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Another way to do this (maybe even easier) is to use the discriminant of the quadratic<BR>\n" );
document.write( "formula.  The quadratic formula applies to all equations of the general form:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2+%2B+bx+%2B+c+=+0 ALIGN=MIDDLE  ALT=\"ax%5E2+%2B+bx+%2B+c+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "The discriminant of this equation is given by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2+-+4%2Aa%2Ac ALIGN=MIDDLE  ALT=\"b%5E2+-+4%2Aa%2Ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >. If the discriminant<BR>\n" );
document.write( "is equal to zero, the trinomial is a perfect square (the equation has two real and <BR>\n" );
document.write( "equal roots).<BR>\n" );
document.write( ".<BR>\n" );
document.write( "By comparing your problem with the general form of the quadratic equation you can see<BR>\n" );
document.write( "that a = 1, b = -24, and c = +48.  Substituting these values into the discriminant,<BR>\n" );
document.write( "you get:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-24%29%5E2-+4%2A1%2A48 ALIGN=MIDDLE  ALT=\"%28-24%29%5E2-+4%2A1%2A48\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "This simplifies to:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=576+-+4%2A48+=+576+-192+=+384 ALIGN=MIDDLE  ALT=\"576+-+4%2A48+=+576+-192+=+384\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Since the discriminant does not equal zero, the given trinomial is not a perfect square.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "However, suppose that the given trinomial had been <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+24x+%2B+144 ALIGN=MIDDLE  ALT=\"x%5E2+-+24x+%2B+144\"   DISABLED_style= \"max-width:90%; height:auto;\" >. In this case<BR>\n" );
document.write( "a = 1, b = -24, and c = 144.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Substitute these values into the discriminant and you get:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2+-+4%2Aa%2Ac+=+%28-24%29%5E2+-+4%2A1%2A144+=+576+-+576+=+0 ALIGN=MIDDLE  ALT=\"b%5E2+-+4%2Aa%2Ac+=+%28-24%29%5E2+-+4%2A1%2A144+=+576+-+576+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "And since the discriminant now equals zero, this trinomial is a perfect square. [As we<BR>\n" );
document.write( "noted earlier the trinomial <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-24x%2B144 ALIGN=MIDDLE  ALT=\"x%5E2+-24x%2B144\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-12%29%5E2 ALIGN=MIDDLE  ALT=\"%28x-12%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.]<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Hope these couple of ways help you to understand your problem better. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );