# Why does Newton's method approximate $\sqrt{z}$ to be $z - \frac{z^2 - x}{2z}$?

According to a document I am reading, I can approximate the square root of some real number $x$ via "Newton's method" by repeatedly taking $z = z - \frac{z^2 - x}{2z}$ after beginning with some random value $z$. This approximation seems pretty good.

However, according to Wikipedia, shouldn't this approximation be $z = z - \frac{z^2}{0.5z^{-0.5}}$ instead? Where did the above equation ($z = z - \frac{z^2 - x}{2z}$) come from?

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It should be $\sqrt x$ in the title, not $\sqrt z$, and all your equations shouldn't be equations; they would make sense either if you replace $=$ by $\to$ or if you replace $z$ by $z_{n+1}$ on the left and by $z_n$ on the right. Finally, I don't see how you derived the prescription you're suggesting; for this to be an application of Newton's method, the denominator would have to be the derivative of the numerator, which it isn't. – joriki Dec 23 '12 at 10:23

Because you use Newton's method on the equation $z^2-x=0$. We don't use an equation that already has square roots in it because we wouldn't know how to compute square roots if we had to use this formula in the first place.