document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=528599>Question 528599</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The length of a rectangle is 3x + 1, its area is 6x^2 + 5x + 1. What in terms of x is its perimeter?<BR>\n" );
document.write( "The width of a rectangle is x - 1, its perimeter is 4x + 6. What in terms of x is its area?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #349346 by </B><A HREF=/tutors/aboutme.mpl?userid=lmeeks54><B>lmeeks54(111)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=lmeeks54><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=lmeeks54><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=349346\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This is actually two problems.  <BR>\n" );
document.write( "...<BR>\n" );
document.write( "First one:<BR>\n" );
document.write( "...<BR>\n" );
document.write( "Let A = area, which is short side (width) x long side (length)<BR>\n" );
document.write( "Given, A = 6x^2 + 5x + 1<BR>\n" );
document.write( "Given one side = 3x + 1  (we don't know if this the width or length)<BR>\n" );
document.write( "...<BR>\n" );
document.write( "If A = side x side, and we are given A and one side, we can divide A by the given side to derive the remaining (unknown) side:<BR>\n" );
document.write( "...<BR>\n" );
document.write( "Since A is in the form of a quadratic equation, if we factor it, the two factors are the equations in terms of x for each of the two sides:<BR>\n" );
document.write( "...<BR>\n" );
document.write( "(3x + 1) (?x + ??) = 6x^2 + 5x + 1<BR>\n" );
document.write( "(3x + 1) (2x + 1) are the two factors and thus, are the two sides.  The missing side we needed to solve for is:<BR>\n" );
document.write( "...<BR>\n" );
document.write( "2x + 1<BR>\n" );
document.write( "...<BR>\n" );
document.write( "However, we are asked for its perimeter in terms of x.  The equation for P, the perimeter, is:<BR>\n" );
document.write( "P = 2 x width + 2 x length<BR>\n" );
document.write( "...<BR>\n" );
document.write( "We already determined the width = 2x + 1<BR>\n" );
document.write( "We were given length = 3x + 1<BR>\n" );
document.write( "...<BR>\n" );
document.write( "P = 2 (2x + 1) + 2 (3x + 1)<BR>\n" );
document.write( "P = 10x + 4<BR>\n" );
document.write( "_ _ _ _ _ _ _ _ _ _ _<BR>\n" );
document.write( "Second problem:<BR>\n" );
document.write( "...<BR>\n" );
document.write( "Given: width = x - 1<BR>\n" );
document.write( "P = 4x + 6<BR>\n" );
document.write( "What is area, A?<BR>\n" );
document.write( "...<BR>\n" );
document.write( "We need to again determine the length of the other side (in this case, the Length), in terms of x<BR>\n" );
document.write( "....<BR>\n" );
document.write( "P = 2 w + 2 L<BR>\n" );
document.write( "P = 2 (x - 1) + 2 ( L) = 4x + 6<BR>\n" );
document.write( "...<BR>\n" );
document.write( "subtract 2x - 2 from both sides:<BR>\n" );
document.write( "2 (L) = 4x + 6 - (2x -2)<BR>\n" );
document.write( "2 (L) = 2x + 8<BR>\n" );
document.write( "...<BR>\n" );
document.write( "divide both sides by 2:<BR>\n" );
document.write( "L = x + 4<BR>\n" );
document.write( "...<BR>\n" );
document.write( "Now we know both sides:<BR>\n" );
document.write( "w = x - 1 (given)<BR>\n" );
document.write( "L = x + 4 (calculated)<BR>\n" );
document.write( "...<BR>\n" );
document.write( "To determine the Area, A, we just multiply the two sides together:<BR>\n" );
document.write( "A = (x - 1) (x + 4)<BR>\n" );
document.write( "A = x^2 - 1x + 4x - 4<BR>\n" );
document.write( "A = X^2 + 3x - 4<BR>\n" );
document.write( "...<BR>\n" );
document.write( "cheers,<BR>\n" );
document.write( "Lee<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );