document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=77193>Question 77193</A>: <I><SPAN STYLE=\"word-break: break-all;\"> For the equation x-2sqrtx=0 , perform the following:<BR>\n" );
document.write( "a) 	Solve for all values of x that satisfies the equation.<BR>\n" );
document.write( "Answer:  <BR>\n" );
document.write( "Show work in this space.<BR>\n" );
document.write( " \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "b) 	Graph the functions y = x and   on the same graph (by plotting points if necessary). Show the points of intersection of these two graphs.<BR>\n" );
document.write( "Graph:<BR>\n" );
document.write( " <BR>\n" );
document.write( "Points of intersection:  <BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #55311 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=55311\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> First part of the problem:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Given:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+-+2%2Asqrt%28x%29+=+0 ALIGN=MIDDLE  ALT=\"x+-+2%2Asqrt%28x%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Find all the values of x that satisfy this equation.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "The method involves getting the radical term on one side of the equation and the other term<BR>\n" );
document.write( "on the other side.  We can do this by adding <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Asqrt%28x%29 ALIGN=MIDDLE  ALT=\"2%2Asqrt%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to both sides and the equation<BR>\n" );
document.write( "then becomes:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+2%2Asqrt%28x%29 ALIGN=MIDDLE  ALT=\"x+=+2%2Asqrt%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Next square both sides. Note that when you do that, the right side squares as follows:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2Asqrt%28x%29%29%5E2+=+%282%5E2%29%2A%28sqrt%28x%29%29%5E2+=+4%2Ax ALIGN=MIDDLE  ALT=\"%282%2Asqrt%28x%29%29%5E2+=+%282%5E2%29%2A%28sqrt%28x%29%29%5E2+=+4%2Ax\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "and the equation therefore squares to:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+=+4x ALIGN=MIDDLE  ALT=\"x%5E2+=+4x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Subtract 4x from both sides and the equation is then:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+4x+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+-+4x+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Factor the common \"x\" from both terms on the left side to get:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2A%28x+-+4%29+=+0 ALIGN=MIDDLE  ALT=\"x%2A%28x+-+4%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Note that this equation will be true if either of the factors is zero.  Therefore,<BR>\n" );
document.write( "one at a time we can set the factors equal to zero to solve the equation.  First:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+0 ALIGN=MIDDLE  ALT=\"x+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "That is one answer. Then:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-4+=+0 ALIGN=MIDDLE  ALT=\"x-4+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Solve by adding 4 to both sides to get:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+4 ALIGN=MIDDLE  ALT=\"x+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "So the answer to the first part of this problem is that the equation given in the problem<BR>\n" );
document.write( "will be satisfied by two values of x ... x = 0 and x = 4.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "For the second part of the problem the two graphs should look like this:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28600%2C400%2C-1%2C20%2C-5%2C+20%2C+x%2C+x+-+2%2Asqrt%28x%29%29 ALIGN=MIDDLE  ALT=\"graph%28600%2C400%2C-1%2C20%2C-5%2C+20%2C+x%2C+x+-+2%2Asqrt%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Note that the original equation given in the problem does not permit x to be less than<BR>\n" );
document.write( "zero because for real values you are not permitted to take the square root of a negative<BR>\n" );
document.write( "number.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "The red line is the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+x ALIGN=MIDDLE  ALT=\"y+=+x\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the green line is the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+x+-+2%2Asqrt%28x%29 ALIGN=MIDDLE  ALT=\"y+=+x+-+2%2Asqrt%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Note that the green line is on the x-axis at the two values of x that we found previously,<BR>\n" );
document.write( "x = 0 and x = +4.  Also note that the only place the two equations intersect is at the<BR>\n" );
document.write( "origin. You can also show this is true mathematically by solving the equation set:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+x ALIGN=MIDDLE  ALT=\"y+=+x\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+x+-+2%2Asqrt%28x%29 ALIGN=MIDDLE  ALT=\"y+=+x+-+2%2Asqrt%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Solve by substitution.  The first equation tells us that y equals x so we can substitute<BR>\n" );
document.write( "x for y in the second equation.  With this substitution, the second equation becomes:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+x+-+2%2Asqrt%28x%29 ALIGN=MIDDLE  ALT=\"x+=+x+-+2%2Asqrt%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Subtract x from both sides and the equation reduces to:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=+-2%2Asqrt%28x%29 ALIGN=MIDDLE  ALT=\"0+=+-2%2Asqrt%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Square both sides and you get:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+0+=+4%2Ax ALIGN=MIDDLE  ALT=\"+0+=+4%2Ax\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "and divide both sides by 4 to find that:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+0 ALIGN=MIDDLE  ALT=\"x+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Then you can solve for the corresponding value of y by returning to the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+x ALIGN=MIDDLE  ALT=\"y+=+x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and plugging in 0 for x to find that y also = 0.  The only common point on the two graphs<BR>\n" );
document.write( "is (0,0) ... the origin as is shown on the graphs themselves.  <BR>\n" );
document.write( ".<BR>\n" );
document.write( "Hope this helps you to understand the problem and how to go about getting the solution <BR>\n" );
document.write( "to it.<BR>\n" );
document.write( ".<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );