document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1116910>Question 1116910</A>: <I><SPAN STYLE=\"word-break: break-all;\"> a curve is defined in the x-y plane by the y=(x-1)(ax^2 +bx+c) where a band c are constants. The curve torches x-axis at the point where x=1 and the line y=3x+6 at the point (-2,0).      <BR>\n" );
document.write( " 1. find the values of a,b and c.     <BR>\n" );
document.write( " 2. sketch the curve and on the same axes, draw the line y=3x+6, indicating clearly the points of the intersection.    <BR>\n" );
document.write( "3. calculate the area of the finite region bounded by the curve and the line y=3x+6 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #732146 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=732146\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "If the curve touches the x-axis at x=1, then it has a double root there.  Then, since it intersects the line y=3x+6 at (-2,0), the equation for the function is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%28x-1%29%5E2%28x%2B2%29 ALIGN=MIDDLE  ALT=\"a%28x-1%29%5E2%28x%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "where a can be any nonzero constant.<br><BR>\n" );
document.write( "The rest of the problem as shown can't be completed, because there was no information to use to evaluate the constant.<br><BR>\n" );
document.write( "So to continue, I will assume the constant is 1; then the equation of the function is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%29%5E2%28x%2B2%29 ALIGN=MIDDLE  ALT=\"%28x-1%29%5E2%28x%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Since the description of the problem said the equation was <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%29%28ax%5E2%2Bbx%2Bc%29 ALIGN=MIDDLE  ALT=\"%28x-1%29%28ax%5E2%2Bbx%2Bc%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, it must be that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc+=+%28x-1%29%28x%2B2%29+=+x%5E2%2Bx-2 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc+=+%28x-1%29%28x%2B2%29+=+x%5E2%2Bx-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  So the answers for part 1 of the problem are<BR>\n" );
document.write( "a=1; b=1; c=-2<br><BR>\n" );
document.write( "Here is the graph for part 2:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-3%2C3%2C-10%2C20%2C%28x-1%29%5E2%2A%28x%2B2%29%2C3x%2B6%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C400%2C-3%2C3%2C-10%2C20%2C%28x-1%29%5E2%2A%28x%2B2%29%2C3x%2B6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The x values of the points of intersection are 0, 1-sqrt(3), and 1+sqrt(3).  You can find the last two by solving the pair of equations simultaneously, knowing that one of the intersection points is at (-2,0).<br><BR>\n" );
document.write( "I will let you do the ugly integration to find the area of the finite regions between the two curves.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );