document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=842482>Question 842482</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How does one solve this equation?<BR>\n" );
document.write( "(&#8730;10+(3&#8730;x)) = (&#8730;x)\r<BR>\n" );
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document.write( "The radical over the 10 covers the whole equation on the left side of the equals sign.\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 #507567 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=507567\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> if i understand you correctly, your equation is:\r<BR>\n" );
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document.write( "sqrt(10 + 3 * sqrt(x)) = sqrt(x)\r<BR>\n" );
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document.write( "pushing this through the algebra.com equation generator make it look like this:\r<BR>\n" );
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document.write( "start with:<BR>\n" );
document.write( "sqrt(10 + 3 * sqrt(x)) = sqrt(x)<BR>\n" );
document.write( "square both sides of this equation to get:<BR>\n" );
document.write( "10 + 3 * sqrt(x) = x<BR>\n" );
document.write( "subtract 10 from both sides of the equation and subtract 3*sqrt(x) from both sides of the equation to get:<BR>\n" );
document.write( "0 = x - 3*sqrt(x) - 10<BR>\n" );
document.write( "commute this to get:<BR>\n" );
document.write( "x - 3*sqrt(x) - 10 = 0<BR>\n" );
document.write( "this is a quadratic equation in disguised form.<BR>\n" );
document.write( "let y = sqrt(x) and this equation becomes:<BR>\n" );
document.write( "y^2 - 3*y - 10 = 0<BR>\n" );
document.write( "factor this equation to get:<BR>\n" );
document.write( "(y-5) * (y+3) = 0<BR>\n" );
document.write( "solve for y to get:<BR>\n" );
document.write( "y = 5 or y = -3<BR>\n" );
document.write( "replace y with sqrt(x) to get:<BR>\n" );
document.write( "sqrt(x) = 5 or sqrt(x) = -3<BR>\n" );
document.write( "solve for x to get:<BR>\n" );
document.write( "x = 25 or x = 9<BR>\n" );
document.write( "now you need to check to see if these solutions are good by replacing x in the original equation with these values of x to see if the equations hold true.\r<BR>\n" );
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document.write( "first we'll do 25.<BR>\n" );
document.write( "the original equation is:<BR>\n" );
document.write( "sqrt(10 + 3 * sqrt(x)) = sqrt(x)<BR>\n" );
document.write( "replace x with 25 to get:<BR>\n" );
document.write( "sqrt(10 + 3 * sqrt(25)) = sqrt(25)<BR>\n" );
document.write( "simplify to get:<BR>\n" );
document.write( "sqrt(10 + 3*5) = 5<BR>\n" );
document.write( "simplify further to get:<BR>\n" );
document.write( "sqrt(25) = 5<BR>\n" );
document.write( "simplify further to get:<BR>\n" );
document.write( "5 = 5<BR>\n" );
document.write( "the equation is true so x = 25 is a good solution.\r<BR>\n" );
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document.write( "now we'll do 9.<BR>\n" );
document.write( "the original equation is:<BR>\n" );
document.write( "sqrt(10 + 3 * sqrt(x)) = sqrt(x)<BR>\n" );
document.write( "replace x with 9 to get:<BR>\n" );
document.write( "sqrt(10 + 3 * sqrt(9)) = sqrt(9)<BR>\n" );
document.write( "simplify to get:<BR>\n" );
document.write( "sqrt(10 + 3*3) = 3<BR>\n" );
document.write( "simplify to get:<BR>\n" );
document.write( "sqrt(19) = 3<BR>\n" );
document.write( "this is not true, so x = -3 cannot be a good solution.\r<BR>\n" );
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document.write( "the problem is that the square root of (9) is not equal to -3.<BR>\n" );
document.write( "it is equal to 3.<BR>\n" );
document.write( "the square root of a number is always the positive root, unless you are solving for y^2.<BR>\n" );
document.write( "example:<BR>\n" );
document.write( "if your equation is y^2 = 9, then y = +/- sqrt(9) = +/- 3<BR>\n" );
document.write( "sqrt(9) can be + 3 or it can be - 3 because you are solving for y^2.<BR>\n" );
document.write( "if  your equation is y = sqrt(9), however, then y = + 3 only.\r<BR>\n" );
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document.write( "the grpah of your 2 equations is shown below.<BR>\n" );
document.write( "they are equal when the graph of the equations intersect.<BR>\n" );
document.write( "the 2 equations to be graphed are:<BR>\n" );
document.write( "y = sqrt(10 + 3*sqrt(x))<BR>\n" );
document.write( "y = sqrt(x)<BR>\n" );
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document.write( "you can see that the graphs intersect at x = 25 and y = 5 which is the coordinate point of (25,5).\r<BR>\n" );
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