document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=728593>Question 728593</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help\r<BR>\n" );
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document.write( "Solve: (sqrt of 2x - 3) = x - 1\r<BR>\n" );
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document.write( "Solve: (sqrt of 3x - 3) - (sqrt of x) = 1\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854696 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13367)</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=854696\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "You can get good mental exercise by solving these equations using logical reasoning and simple mental arithmetic.<br><BR>\n" );
document.write( "Start by assuming (hoping) that the solutions are integers.  Then use logical reasoning to determine the solution(s).<br><BR>\n" );
document.write( "(1) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282x-3%29=x-1 ALIGN=MIDDLE  ALT=\"sqrt%282x-3%29=x-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "In this equation, we want (2x-3) to be a perfect square. And if x is an integer, the (2x-3) is an odd perfect square.  So look at values of x for which (2x-3) is an odd perfect square.<br><BR>\n" );
document.write( "(a) 2x-3 = 1; 2x = 4; x = 2.  sqrt(2x-3) = sqrt(1) = 1; and 2-1 = 1.  So x = 2 is a solution.<br><BR>\n" );
document.write( "(b) 2x-3 = 9; 2x = 12; x = 6.  sqrt(2x-3) = 3 but x-1 = 6.  x = 6 is not a solution.<br><BR>\n" );
document.write( "For larger values of x, \"x-1\" will increase faster than \"sqrt(2x-3)\", so there are no more solutions.<br><BR>\n" );
document.write( "ANSWER: x = 2<br><BR>\n" );
document.write( "(2) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%283x-3%29-sqrt%28x%29=1 ALIGN=MIDDLE  ALT=\"sqrt%283x-3%29-sqrt%28x%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "In this equation, we want x and (3x-3) to both be perfect squares.<br><BR>\n" );
document.write( "(a) x = 1; 3x-3 = 0; both are perfect squares.  However, trying x = 1 in the equation gives us \"0-1 = 1\", which is not true.  Note that the two sides of the equation evaluate to numbers that are opposites.  This is a common occurrence in equations like this, where the formal solution process including squaring both sides of an equation leads to extraneous solutions.<br><BR>\n" );
document.write( "So x = 1 is NOT a solution.<br><BR>\n" );
document.write( "(b) x = 4; 3x-3 = 9 is also a perfect square; and sqrt(3x-3) = 3 and x-1 = 3, so x = 4 is a solution.<br><BR>\n" );
document.write( "Similarly to the first problem, for larger values of x \"sqrt(3x-3)\" will increase faster than \"sqrt(x)\", so again there are no more solutions.<br><BR>\n" );
document.write( "ANSWER: x = 4<br><BR>\n" );
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