document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=177767>Question 177767</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please solve \r<BR>\n" );
document.write( "\n" );
document.write( "A. (3-&#8730;(x+2) )^2=4\r<BR>\n" );
document.write( "\n" );
document.write( "and\r<BR>\n" );
document.write( "\n" );
document.write( "B. 3+5&#8730;(1+2x ^2 )=13\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "thank you! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #132796 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=132796\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <FONT FACE=\"Times New Roman\" SIZE=\"+2\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "A.  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large (3 - sqrt{x+2})^2 = 4\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<FONT COLOR=#FF000> Step 1: <FONT COLOR=#00000>Expand the binomial on the left:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{        } \math 9 - 6 sqrt{x + 2} + (x + 2) = 4\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<FONT COLOR=#FF000>Step 2:<FONT COLOR=#00000> A simple substitution will allow us to deal with this ugliness in a fairly straightforward manner.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large t = sqrt{x + 2}\">, then:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{        } \math 9 - 6t + t^2 = 4\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<FONT COLOR=#FF000>Step 3:<FONT COLOR=#00000> Put the quadratic into standard form:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{        } \math t^2 - 6t + 5 = 0\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Giving us a factorable quadratic in <i>t</i>, so:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<FONT COLOR=#FF000>Step 4:<FONT COLOR=#00000> Factor the quadratic:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{        } \math (t - 1)(t - 5) = 0\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Therefore <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{    } \math  t = 1 \text{     }\"> or <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{    } \math t = 5 \text{     }\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "But we aren't done yet.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<FONT COLOR=#FF000>Step 5:<FONT COLOR=#00000> Since <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large t = sqrt{x + 2}\">,\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{        } \math sqrt{x + 2} = 1\"><IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{   or        } \math sqrt{x + 2} = 5\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{        } \math x + 2 = 1\"><IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{   or        } \math x + 2 = 25\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{        } \math x = -1\"><IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{   or        } \math x = 23\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "But we are <i>still</i> not done.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<FONT COLOR=#FF000>Step 6:<FONT COLOR=#00000>  Check the potential roots to ensure that we have not introduced extraneous roots in the process of solving the problem:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x = -1\"> then,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{        } \math (3 - sqrt{-1+2})^2 = 4\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{        } \math (3 - 1)^2 = 4\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{        } \math (2)^2 = 4\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<FONT COLOR=#00C00>True statement, potential solution <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x = -1\"> confirmed.<FONT COLOR=#00000>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x = 23\"> then,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{        } \math (3 - sqrt{23+2})^2 = 4\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{        } \math (3 - sqrt{25})^2 = 4\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{        } \math (3 - 5)^2 = 4\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{        } \math (-2)^2 = 4\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<FONT COLOR=#00C00>True statement, potential solution <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x = 23\"> confirmed.<FONT COLOR=#00000>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<i>Now</i> we are done.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "=======================================================================\r<BR>\n" );
document.write( "\n" );
document.write( "B.  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 3 + 5 sqrt{1 + 2x^2} = 13\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<FONT COLOR=#FF000>Step 1:<FONT COLOR=#00000> Add -3 to both sides and then multiply by <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large \frac {1}{5}\"> to obtain:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{        } \math sqrt{1 + 2x^2} = 2\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<FONT COLOR=#FF000>Step 2:<FONT COLOR=#00000> Square both sides:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{        } \math 1 + 2x^2 = 4\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<FONT COLOR=#FF000>Step 3:<FONT COLOR=#00000> Add -1 to both sides and then multiply by <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large \frac {1}{2}\"> to obtain:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{        } \math x^2 = \frac {3}{2}\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<FONT COLOR=#FF000>Step 4:<FONT COLOR=#00000> Take the square root of both sides, considering both the positive and negative roots, and then rationalize the denominators:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{        } \math x = \pm sqrt{\frac {3}{2}}\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{        } \math x = \pm \frac {sqrt{6}}{2}\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<FONT COLOR=#FF000>Step 5:<FONT COLOR=#00000> Check the potential solutions.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x^2 = \frac {3}{2}\"> regardless of whether <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x =  \frac {sqrt{6}}{2}\"> or <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x =  -\frac {sqrt{6}}{2}\">, if one solution is good, then they both are:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{        } \math 3 + 5 sqrt{1 + 2 (\frac {3}{2})} = 13\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{        } \math 3 + 5 sqrt{1 + 3} = 13\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{        } \math 3 + 10 = 13\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<FONT COLOR=#00C00>True statement, both potential solutions <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x = \pm \frac{sqrt{6}}{2}\"> confirmed.<FONT COLOR=#00000><BR>\n" );
document.write( "</FONT> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );