document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=717301>Question 717301</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the extraneous solution of sqrt(x-3)= (x-5)<BR>\n" );
document.write( "also sqrt of(x+4)=sqrt(3x) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #440232 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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=440232\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> First we find solutions. Then we determine which, if any, are extraneous.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x-3%29=+%28x-5%29 ALIGN=MIDDLE  ALT=\"sqrt%28x-3%29=+%28x-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Square both sides:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%28x-3%29%29%5E2=+%28x-5%29%5E2 ALIGN=MIDDLE  ALT=\"%28sqrt%28x-3%29%29%5E2=+%28x-5%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Squaring the left side is as easy as it looks. Squaring an expression like the right side is where a lot of mistakes are made. It is <u><n>not</b></u> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-5%5E2 ALIGN=MIDDLE  ALT=\"x%5E2-5%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >!! To square a two-term expression like (x-5) correctly we must use FOIL on (x-5)(x-5) or use the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a-b%29%5E2+=+a%5E2-2ab%2Bb%5E2 ALIGN=MIDDLE  ALT=\"%28a-b%29%5E2+=+a%5E2-2ab%2Bb%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > pattern. I prefer using the pattern:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-3=+%28x%29%5E2-2%28x%29%285%29%2B5%5E2 ALIGN=MIDDLE  ALT=\"x-3=+%28x%29%5E2-2%28x%29%285%29%2B5%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which simplifies to:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-3=+x%5E2-10x%2B25 ALIGN=MIDDLE  ALT=\"x-3=+x%5E2-10x%2B25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now that x is out of the square root we can solve for it. This is a quadratic equation so we want one side to be zero. Subtracting x and adding 3 we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=+x%5E2-11x%2B28 ALIGN=MIDDLE  ALT=\"0+=+x%5E2-11x%2B28\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Next we factor (or use the Quadratic Formula). This factors fairly easily:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=+%28x-7%29%28x-4%29 ALIGN=MIDDLE  ALT=\"0+=+%28x-7%29%28x-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "From the Zero Product Property:<BR>\n" );
document.write( "x-7 = 0 or x-4 = 0<BR>\n" );
document.write( "Solving these we get:<BR>\n" );
document.write( "x = 7 or x = 4<br><BR>\n" );
document.write( "Now we check. Use the original equation:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x-3%29=+%28x-5%29 ALIGN=MIDDLE  ALT=\"sqrt%28x-3%29=+%28x-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Checking x = 7:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%287-3%29=+%287-5%29 ALIGN=MIDDLE  ALT=\"sqrt%287-3%29=+%287-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%284%29=+%282%29 ALIGN=MIDDLE  ALT=\"sqrt%284%29=+%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2+=+%282%29 ALIGN=MIDDLE  ALT=\"2+=+%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Check!<BR>\n" );
document.write( "Checking x = 4:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%284-3%29=+%284-5%29 ALIGN=MIDDLE  ALT=\"sqrt%284-3%29=+%284-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%281%29+=+%28-1%29 ALIGN=MIDDLE  ALT=\"sqrt%281%29+=+%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+=+%28-1%29 ALIGN=MIDDLE  ALT=\"1+=+%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Check fails!<BR>\n" );
document.write( "So x = 7 is actually a solution and x = 4 is extraneous.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%2B4%29=sqrt%283x%29 ALIGN=MIDDLE  ALT=\"sqrt%28x%2B4%29=sqrt%283x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Square both sides:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%28x%2B4%29%29%5E2=%28sqrt%283x%29%29%5E2 ALIGN=MIDDLE  ALT=\"%28sqrt%28x%2B4%29%29%5E2=%28sqrt%283x%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "x+4 = 3x<BR>\n" );
document.write( "Subtract x:<BR>\n" );
document.write( "4 = 2x<BR>\n" );
document.write( "Divide by 2:<BR>\n" );
document.write( "2 = x<BR>\n" );
document.write( "Check.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%2B4%29=sqrt%283x%29 ALIGN=MIDDLE  ALT=\"sqrt%28x%2B4%29=sqrt%283x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Checking x = 2:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282%2B4%29=sqrt%283%282%29%29 ALIGN=MIDDLE  ALT=\"sqrt%282%2B4%29=sqrt%283%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%286%29=sqrt%286%29 ALIGN=MIDDLE  ALT=\"sqrt%286%29=sqrt%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Check!<BR>\n" );
document.write( "x = 2 is the solution to this equation. There are no extraneous solutions.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );