In this particular case, the iterates oscillate between 1 and -1, so the method fails to converge. As others here have noted, it is doomed to fail as long as the initial estimate x0 is real, since the two solutions are both complex (imaginary, in fact).
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In this particular case, the iterates oscillate between 1 and -1, so the method fails to converge. As others here have noted, it is doomed to fail as long as the initial estimate x0 is real, since the two solutions are both complex (imaginary, in fact).
Did you plug it into Newton's method and see? If not, why not?
x1 = x0 - f(x0)/f'(x0)
x2 = x1 - f(x1)/f'(x1)
x3 = x2 - f(x2)/f'(x2)
etc.
Just keep plugging the value of x in to get the next value of x. Did you do this even once?
f(x) = x^2 + 3
f'(x) = 2x
Newton's method is only applicable when the roots are real numbers, or at least one root is real.
In your case both roots are imaginary, so the method fails.
The method will fail to converge.