Newtons Method by Recursive Quines
Common Lisp, 223 95 68 66
(#1=(lambda(x p)(format t&quot;~S~%~S&quot;p`(,x',x,(+(/ p 2)(/ p)))))'#1#1)

Now that I read the problem statement more carefully (thanks, primo!) I noticed that the first line must be the result of calculation, not that it needs to contain the result. Thus, I think my earlier attempts did not quite follow the rules. This one should.
Example use (SBCL 1.1.15):
$ sbcl --script nq.lisp | tee nq2.lisp
1
((LAMBDA (X P) (FORMAT T &quot;~S~%~S&quot; P `(,X ',X ,(+ (/ P 2) (/ P)))))
 '(LAMBDA (X P) (FORMAT T &quot;~S~%~S&quot; P `(,X ',X ,(+ (/ P 2) (/ P))))) 3/2)
$ sbcl --script nq2.lisp | tee nq3.lisp
3/2
((LAMBDA (X P) (FORMAT T &quot;~S~%~S&quot; P `(,X ',X ,(+ (/ P 2) (/ P)))))
 '(LAMBDA (X P) (FORMAT T &quot;~S~%~S&quot; P `(,X ',X ,(+ (/ P 2) (/ P))))) 17/12)
$ sbcl --script nq3.lisp | tee nq4.lisp
17/12
((LAMBDA (X P) (FORMAT T &quot;~S~%~S&quot; P `(,X ',X ,(+ (/ P 2) (/ P)))))
 '(LAMBDA (X P) (FORMAT T &quot;~S~%~S&quot; P `(,X ',X ,(+ (/ P 2) (/ P))))) 577/408)
$ sbcl --script nq4.lisp | tee nq5.lisp
577/408
((LAMBDA (X P) (FORMAT T &quot;~S~%~S&quot; P `(,X ',X ,(+ (/ P 2) (/ P)))))
 '(LAMBDA (X P) (FORMAT T &quot;~S~%~S&quot; P `(,X ',X ,(+ (/ P 2) (/ P)))))
 665857/470832)
$ sbcl --script nq5.lisp | tee nq6.lisp
665857/470832
((LAMBDA (X P) (FORMAT T &quot;~S~%~S&quot; P `(,X ',X ,(+ (/ P 2) (/ P)))))
 '(LAMBDA (X P) (FORMAT T &quot;~S~%~S&quot; P `(,X ',X ,(+ (/ P 2) (/ P)))))
 886731088897/627013566048)
$