document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=477556>Question 477556</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A square has an area of x^2 + 6x + 9 find the length of a side. Make a sketch of the square Please help </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #327296 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=327296\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the area of a square is equal to s*s = s^2<BR>\n" );
document.write( "s = 1 side of the square.<BR>\n" );
document.write( "the area of the square is equal to x^2 + 6x + 9<BR>\n" );
document.write( "this means that s^2 = x^2 + 6x + 9<BR>\n" );
document.write( "this means that s = sqrt(x^2 + 6x + 9)<BR>\n" );
document.write( "it just so happens that x^2 + 6x + 9 is a perfect square.<BR>\n" );
document.write( "it is equal to (x+3)^2<BR>\n" );
document.write( "this means that s^2 = (x+3)^2 which means that:<BR>\n" );
document.write( "sqrt(s^2) = +/- sqrt(x+3)^2 which means that:<BR>\n" );
document.write( "s = +/- (x+3)<BR>\n" );
document.write( "if s = + (x+3) then:<BR>\n" );
document.write( "s = x + 3<BR>\n" );
document.write( "if s = -(x+3) then:<BR>\n" );
document.write( "s = -x - 3<BR>\n" );
document.write( "now s has to be > 0.<BR>\n" );
document.write( "this means that:<BR>\n" );
document.write( "x+3 > 0 which means that x > -3<BR>\n" );
document.write( "this also means that:<BR>\n" );
document.write( "-x-3 > 0 which means that x < -2<BR>\n" );
document.write( "this means that x can be any real number except -3.<BR>\n" );
document.write( "when x is equal to -3, s is equal to 0 and the area of the square is equal to 0 which means that you don't have a square of any magnitude.<BR>\n" );
document.write( "the length of each side of your square is dependent on the value of x.<BR>\n" );
document.write( "a graph of the equation of x^2 + 6x + 9 is shown below:<BR>\n" );
document.write( "this graph shows the area of the square based on the value of x.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-10%2C10%2C-10%2C10%2Cx%5E2+%2B+6x+%2B+9%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-10%2C10%2C-10%2C10%2Cx%5E2+%2B+6x+%2B+9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "regardless of the value of x, this graph will be positive which means the area of the square will be positive.<BR>\n" );
document.write( "this is everywhere except when x = -3.<BR>\n" );
document.write( "when x = -3, the area is zero which we already indicated is not valid.<BR>\n" );
document.write( "each side of your square, however, is not fixed to any one length.<BR>\n" );
document.write( "the length of each side is dependent on the value of x.<BR>\n" );
document.write( "a graph of the equation of y = +/- sqrt(x^2 + 6x + 9)}}} is shown below:<BR>\n" );
document.write( "this graph is showing the value of the side of the square based on the value of x.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-10%2C10%2C-10%2C10%2Csqrt%28x%5E2+%2B+6x+%2B+9%29%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-10%2C10%2C-10%2C10%2Csqrt%28x%5E2+%2B+6x+%2B+9%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "you can see that the length of each side will always be positive.<BR>\n" );
document.write( "it's low point is 0 when x = -3, which is disallowed.<BR>\n" );
document.write( "you asked me to draw your square.<BR>\n" );
document.write( "any square will do.<BR>\n" );
document.write( "just draw a rectangle which has all sides equal to each other.<BR>\n" );
document.write( "each side of this square will be equal to sqrt(x^2 + 6x + 9).<BR>\n" );
document.write( "x can be any value except -3.<BR>\n" );
document.write( "you can also show the value of each side of the square as follows:<BR>\n" );
document.write( "s = absolute value of (x+3).<BR>\n" );
document.write( "in formula terms, this looks like:<BR>\n" );
document.write( "s = |x+3|<BR>\n" );
document.write( "when (x+3) is positive, you get s = x + 3<BR>\n" );
document.write( "when (x+3) is negative, you get s = -(x + 3) which becomes s = -x - 3.<BR>\n" );
document.write( "these are the same values we calculated above.<BR>\n" );
document.write( "as ugly as it might look, that's your answer.<BR>\n" );
document.write( "here's a picture of your square:<BR>\n" );
document.write( "the length of each side in the square is equal to |x+3|.<BR>\n" );
document.write( "this means each side in the square is equal to the absolute value of (x+3).<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/problems/square_1.jpg\" alt = \"$$$$$\"/ ><BR>\n" );
document.write( "an example:<BR>\n" );
document.write( "let x = -5.<BR>\n" );
document.write( "each side of your square is equal to |-5+3| which is equal to |-2| which is equal to 2.<BR>\n" );
document.write( "the area of your square is equal to x^2 + 6x + 9 after you replace x with (-5) to get (-5)^2 + 6*(-5) + 9 which is equal to 25 - 30 + 9 which is equal to 4.<BR>\n" );
document.write( "since 2^4 = 4, the dimensions of each side and the area of the square confirm that the value of x = |-2| is good.<BR>\n" );
document.write( "s is equal to 2<BR>\n" );
document.write( "s^2 is equal to 4\r<BR>\n" );
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document.write( " </SPAN>\n" );
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