document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=372650>Question 372650</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the solution for this square root equation?\r<BR>\n" );
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document.write( "sqrt (5x-5) - sqrt (4x-3 =1)\r<BR>\n" );
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document.write( "I am not sure how to solve this problem as i keep coming up with multiple answers. Please Help. 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 #265443 by </B><A HREF=/tutors/aboutme.mpl?userid=Jk22><B>Jk22(389)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Jk22><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=Jk22><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=265443\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  <BR>\n" );
document.write( "Sensible points : <BR>\n" );
document.write( "*) Definition of the set of x<BR>\n" );
document.write( "*) Definition of Sqrt. (function or multi-valuate)\r<BR>\n" );
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document.write( "Supposition : x is in the set of real numbers, only real image allowed.<BR>\n" );
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document.write( "sqrt (5x-5) - sqrt (4x-3) = 1<BR>\n" );
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document.write( "Guessing : 1 is not solution (-1<>1), except if Sqrt(1)=+/- 1. 6 and 3 leaves not exact roots.<BR>\n" );
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document.write( "  => solution 1, if sqrt(1)=-1 (sqrt negatively defined)<BR>\n" );
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document.write( "Problem by blind Equation manipulation : most probable steps <BR>\n" );
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document.write( "1) root elimination : <BR>\n" );
document.write( " 1a) squaring<BR>\n" );
document.write( " 1b) remaining root isolation<BR>\n" );
document.write( " 1c) re-squaring -> 2nd degree equation<BR>\n" );
document.write( "2) Verification <BR>\n" );
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document.write( "1a) 5x-5 + 4x-3 -2Sqrt((5x-5)(4x-3)) = 1<BR>\n" );
document.write( "1b) -2Sqrt((5x-5)(4x-3)) = 1 - 9x + 8<BR>\n" );
document.write( "1c) 4(5x-5)(4x-3) = 9(1-x) = 20(x-1)(4x-3)<BR>\n" );
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document.write( "solution if x = 1 ? Not by the guessing point if sqrt is uni-valuate positive.<BR>\n" );
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document.write( "suppose x<>1, divide by x-1 : <BR>\n" );
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document.write( "20(4x-3) = -9 = 80x - 60<BR>\n" );
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document.write( "80x = 51<BR>\n" );
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document.write( "x = 51/80<BR>\n" );
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document.write( "But x-1<0, hence 51/80 is not a solution since the roots are either real or imaginary.\r<BR>\n" );
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document.write( "Supposition 2 : x are complex numbers : x = a + ib<BR>\n" );
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document.write( "let sqrt(5x-5) = p+iq => \r<BR>\n" );
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document.write( "1) p^2 - q^2 = 5a - 5<BR>\n" );
document.write( "2) 2pq = 5b<BR>\n" );
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document.write( "sqrt(4x-3) = s + it =><BR>\n" );
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document.write( "3) s^2 -t^2 = 4a - 3<BR>\n" );
document.write( "4) 2st = 5b<BR>\n" );
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document.write( "and p-s = 1, q+t = 0\r<BR>\n" );
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document.write( "s=p-1, t = -q, get : (2)&(4) : 2(p-1)q = -4b = 2pq - 2q = 5b - 2q<BR>\n" );
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document.write( "q = 9b/2<BR>\n" );
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document.write( "then (2) : 2p9b/2 = 5b, if b<>0, then p = 5/9<BR>\n" );
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document.write( "equ (1) and (3) : <BR>\n" );
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document.write( "5((p-1)^2-q^2+3) = 4(p^2 - q^2 + 5)<BR>\n" );
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document.write( "p^2 - q^2 = 10p<BR>\n" );
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document.write( "q^2 = p^2 - 10p < 0, hence there are no solution more in the complex numbers.<BR>\n" );
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document.write( "3rd possibility remains if x is a matrix.\r<BR>\n" );
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