Recently I was introduced to the Newton-Raphson method for finding roots of a polynomial function. I looked up the proof of it and I found this.
I found a variation of the Newton-Raphson method by considering the first $3$ terms of the Taylor series. i.e, in
$$f(\alpha) = f(x) + (\alpha - x)f'(x) + (\alpha-x)^2\frac{f''(x)}{2!} +\;\; ...$$ We consider
$$f(\alpha) \approx f(x) + (\alpha - x)f'(x) + (\alpha-x)^2\frac{f''(x)}{2!}$$
And as $f(\alpha) = 0$, the above equation turns to
$$0 \approx f(x) + (\alpha - x)f'(x) + (\alpha-x)^2\frac{f''(x)}{2!}$$
And then we solve the above equation for $\alpha$ by using the quadratic formula.
So, my question is will this variation have a worse time complexity for finding the roots of a polynomial function than the original Newton-Raphson method?