document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208758>Question 1208758</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find an approximation for squareRoot17 without a calculator. Round to nearest thousandth. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #847191 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=847191\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "To approximate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28R%29 ALIGN=MIDDLE  ALT=\"sqrt%28R%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we can use this equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+0.5%2A%28x%2BR%2Fx%29 ALIGN=MIDDLE  ALT=\"y+=+0.5%2A%28x%2BR%2Fx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "In this case our radicand is R = 17, so the equation will update to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+0.5%2A%28x%2B17%2Fx%29 ALIGN=MIDDLE  ALT=\"y+=+0.5%2A%28x%2B17%2Fx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "See <a  rel=nofollow HREF=\"https://en.wikipedia.org/wiki/Methods_of_computing_square_roots\">this page</a> for more info. In that link, scroll down until you reach the subheading labeled \"Heron's Method\". This is not to be confused with the Heron triangle area formula.\r<BR>\n" );
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document.write( "The initial x value is a guess.<BR>\n" );
document.write( "Let's say we pick x = 4.5 since it's between 4 and 5 (note that 4^2 = 16 and 5^2 = 25; and 16 < 17 < 25)<BR>\n" );
document.write( "We feed that guess into the equation. The result is then the next guess to try. <BR>\n" );
document.write( "The process keeps going until both columns agree on the same decimal digits.\r<BR>\n" );
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document.write( "Table of values<BR>\n" );
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document.write( "<tr><TD >x</TD><TD >y = 0.5*(x+17/x)</TD></tr><tr><TD >4.5</TD><TD >4.13888888888889</TD></tr><tr><TD >4.13888888888889</TD><TD >4.12313571961223</TD></tr><tr><TD ><font color=red>4.1231</font>3571961223</TD><TD ><font color=red>4.1231</font>0562572749</TD></tr>\n" );
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document.write( "I have highlighted in red where the two columns agree on the partial decimal representation. We can confidently conclude that sqrt(17) = <font color=red>4.123</font> when rounding to 3 decimal places.\r<BR>\n" );
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document.write( "Note that<BR>\n" );
document.write( "4.123^2 = 16.999129<BR>\n" );
document.write( "4.1231^2 = 16.99995361<BR>\n" );
document.write( "both of which can be found by hand (given enough time) \r<BR>\n" );
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document.write( "Answer: <font color=red>4.123</font><BR>\n" );
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