document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=616469>Question 616469</A>: <I><SPAN STYLE=\"word-break: break-all;\"> help me solve this please: sqrt(x-3) = (no sqrt anymore) 6-x </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #387712 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=387712\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x-3%29+=+6-x ALIGN=MIDDLE  ALT=\"sqrt%28x-3%29+=+6-x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since the square root is by itself already we can go ahead and 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+=+%286-x%29%5E2 ALIGN=MIDDLE  ALT=\"%28sqrt%28x-3%29%29%5E2+=+%286-x%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Squaring the square root is easy. Squaring the right side can easily be done incorrectly. You must use FOIL on (6-x)(6-x) or use the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a-b%29%5E2+=+a%5E2-2ab%2Bb%5E3 ALIGN=MIDDLE  ALT=\"%28a-b%29%5E2+=+a%5E2-2ab%2Bb%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > pattern. I like using the pattern:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-3+=+%286%29%5E2-2%286%29%28x%29+%2B+%28x%29%5E2 ALIGN=MIDDLE  ALT=\"x-3+=+%286%29%5E2-2%286%29%28x%29+%2B+%28x%29%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%5E2-12x+%2B+36 ALIGN=MIDDLE  ALT=\"x-3+=+x%5E2-12x+%2B+36\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now we have an equation we can solve. It 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+-13x+%2B+39 ALIGN=MIDDLE  ALT=\"0+=+x%5E2+-13x+%2B+39\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Next we factor or use the Quadratic Formula. This won't factor so we have to use the formula:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-%28-13%29+%2B-+sqrt%28%28-13%29%5E2+-4%281%29%2839%29%29%29%2F2%281%29 ALIGN=MIDDLE  ALT=\"x+=+%28-%28-13%29+%2B-+sqrt%28%28-13%29%5E2+-4%281%29%2839%29%29%29%2F2%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Simplifying...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-%28-13%29+%2B-+sqrt%28169+-4%281%29%2839%29%29%29%2F2%281%29 ALIGN=MIDDLE  ALT=\"x+=+%28-%28-13%29+%2B-+sqrt%28169+-4%281%29%2839%29%29%29%2F2%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-%28-13%29+%2B-+sqrt%28169+-156%29%29%2F2%281%29 ALIGN=MIDDLE  ALT=\"x+=+%28-%28-13%29+%2B-+sqrt%28169+-156%29%29%2F2%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-%28-13%29+%2B-+sqrt%2813%29%29%2F2%281%29 ALIGN=MIDDLE  ALT=\"x+=+%28-%28-13%29+%2B-+sqrt%2813%29%29%2F2%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2813+%2B-+sqrt%2813%29%29%2F2 ALIGN=MIDDLE  ALT=\"x+=+%2813+%2B-+sqrt%2813%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2813+%2B+sqrt%2813%29%29%2F2 ALIGN=MIDDLE  ALT=\"x+=+%2813+%2B+sqrt%2813%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2813+-+sqrt%2813%29%29%2F2 ALIGN=MIDDLE  ALT=\"x+=+%2813+-+sqrt%2813%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Since we squared both sides of the equation, which is not a mistake, we <i>must</i> check out solutions. Use the original equation:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x-3%29+=+6-x ALIGN=MIDDLE  ALT=\"sqrt%28x-3%29+=+6-x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Checking <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2813+%2B+sqrt%2813%29%29%2F2 ALIGN=MIDDLE  ALT=\"x+=+%2813+%2B+sqrt%2813%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%28%2813+%2B+sqrt%2813%29%29%2F2%29-3%29+=+6-%28%2813+%2B+sqrt%2813%29%29%2F2%29 ALIGN=MIDDLE  ALT=\"sqrt%28%28%2813+%2B+sqrt%2813%29%29%2F2%29-3%29+=+6-%28%2813+%2B+sqrt%2813%29%29%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Looking at the right side...<BR>\n" );
document.write( "Since the 6 = 12/2 and since the x value, (13 + sqrt(13))/2, is clearly more than 12/2, the right side turns out negative. The left side is a square root which cannot be negative. So this x value does not check and we reject it. (Note: this answer did not happen because we made a mistake. These \"non-solutiosn\" can happen any time you square both sides of an equation.)<br><BR>\n" );
document.write( "Checking <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2813+-+sqrt%2813%29%29%2F2 ALIGN=MIDDLE  ALT=\"x+=+%2813+-+sqrt%2813%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%28%2813+-+sqrt%2813%29%29%2F2%29-3%29+=+6-%28%2813+-+sqrt%2813%29%29%2F2%29 ALIGN=MIDDLE  ALT=\"sqrt%28%28%2813+-+sqrt%2813%29%29%2F2%29-3%29+=+6-%28%2813+-+sqrt%2813%29%29%2F2%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%28%28%2813+-+sqrt%2813%29%29%2F2%29-6%2F2%29+=+12%2F2-%28%2813+-+sqrt%2813%29%29%2F2%29 ALIGN=MIDDLE  ALT=\"sqrt%28%28%2813+-+sqrt%2813%29%29%2F2%29-6%2F2%29+=+12%2F2-%28%2813+-+sqrt%2813%29%29%2F2%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%28%287+-+sqrt%2813%29%29%2F2%29+=+%28-1+%2B+sqrt%2813%29%29%2F2%29 ALIGN=MIDDLE  ALT=\"sqrt%28%287+-+sqrt%2813%29%29%2F2%29+=+%28-1+%2B+sqrt%2813%29%29%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "With the square root inside the square root, the rest of the check is bit of a mess. Before we square both sides, let's rewrite the right side as a binomial (so it will be easier to square):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%287+-+sqrt%2813%29%29%2F2%29+=+sqrt%2813%29%2F2+-+1%2F2%29 ALIGN=MIDDLE  ALT=\"sqrt%28%287+-+sqrt%2813%29%29%2F2%29+=+sqrt%2813%29%2F2+-+1%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Squaring both sides:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%28%287+-+sqrt%2813%29%29%2F2%29%29%5E2+=+%28sqrt%2813%29%2F2+-+1%2F2%29%5E2%29 ALIGN=MIDDLE  ALT=\"%28sqrt%28%287+-+sqrt%2813%29%29%2F2%29%29%5E2+=+%28sqrt%2813%29%2F2+-+1%2F2%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%287+-+sqrt%2813%29%29%2F2+=+%28sqrt%2813%29%2F2%29%5E2+-+2%28sqrt%2813%29%2F2%29%281%2F2%29+%2B+%281%2F2%29%5E2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2F2+-+sqrt%2813%29%2F2+=+13%2F4+-+sqrt%2813%29%2F2+%2B+1%2F4 ALIGN=MIDDLE  ALT=\"7%2F2+-+sqrt%2813%29%2F2+=+13%2F4+-+sqrt%2813%29%2F2+%2B+1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2F2+-+sqrt%2813%29%2F2+=+14%2F4+-+sqrt%2813%29%2F2 ALIGN=MIDDLE  ALT=\"7%2F2+-+sqrt%2813%29%2F2+=+14%2F4+-+sqrt%2813%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2F2+-+sqrt%2813%29%2F2+=+7%2F2+-+sqrt%2813%29%2F2 ALIGN=MIDDLE  ALT=\"7%2F2+-+sqrt%2813%29%2F2+=+7%2F2+-+sqrt%2813%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Check!<br><BR>\n" );
document.write( "So the only solution to your equation is: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%2813+-+sqrt%2813%29%29%2F2 ALIGN=MIDDLE  ALT=\"x+=+%2813+-+sqrt%2813%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "P.S. Sorry about my earlier typo. I typed a -12 for the \"b\" instead of a -13. When I ended up with complex roots I should have thought to check my typing. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );