Newton's Method derivative calculation

Question

I'm trying to reduce this form of Newtons Method for:

$f(x) = x^2 - a$, $f'(x) = 2x$

$$x_1 = x_0 - \frac {f(x_0)}{f'(x_0)}$$

to the form, which I am told is the correct expanded form:

$(x/2) + (a/(2x))$

I start:

$x_1 = x - (x^2 - a)/(2x)$

$x_1 = x - (x^2)/(2x) + a/(2x)$

$x_1 = 2x/2 - x/2 + a/2x$

$x_1 = x/2 + a/2x$

Unfortunately, I'm unable to see where the signs are incorrect for the last form. Does anyone know where I went wrong in changing the form of this expression? Thanks

UPDATE revised with solution

Answer 1

Multiply the first term in the Newton step by $\frac{2x_0}{2x_0}$ and combine the resulting fractions: \begin{align*} x_1 ={}& x_0 - \frac{x_0^2 - a}{2x_0} \\ ={}& \frac{2x_0^2 - x_0^2 + a}{2x_0} \\ ={}& \frac{x_0^2 + a}{2x_0} \\ ={}& \frac{x_0}{2} + \frac{a}{2x_0}. \end{align*}