Alan

Here's a direct derivation:

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Here's yet another approach:

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Have a look at https://en.wikipedia.org/wiki/Newton%27s_method for a derivation of the Newton-Raphson method.

Applying the technique tp f(x) = x² - 5 we have:

f(x) = x² - 5

f`(x) = 2x

x_n+1 = x_n - (x_n² - 5)/(2x_n)

x_n+1 = (2x_n² - (x_n² - 5))/(2x_n) 

x_n+1 = (2x_n² - x_n² + 5))/(2x_n) 

x_n+1 = (x_n² + 5))/(2x_n) 

x_n+1 = (x_n + 5/x_n)/2 

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If f(x) = 3x - 1 then f(-2) means replace the x by -2 in the expression 3x - 1, and calculate the result.

So here we have f(-2) = 3*(-2) - 1 → -6 - 1 → -7

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This (and your previous similar question) are slightly unclear, as you don't specify what measure is to be used for "size".  If it's the relative areas that you are looking for then that is:  (7*10^5/2.3*10^8)^2 ≈  9.26*10^-6  (note the -6, not one 6th).

If by "size" you mean relative radii, then you don't need to square the ratio.

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I suspect it is the ratio of the areas that is wanted, so  ratio = (7*10^5/1.5*10^8)^2 → (4.667*10^-3)^2 ≈ 2.178*10^-5

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Using Heron's formula the area of an equilateral triangle can be written as:

A = √(s(s-a)^3)  where a is side length and s is semi-perimeter (=3a/2 here)

If A = a numerically. then:

a = √(3a/2(a/2)^3)   or   a = (a/2)^2*√3  so   1 = (a/4)√3    or   a = 4/√3 → (4√3)/3

Hence perimeter = 3a → 4√3

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There are no integer values for SB that satisfy this relationship.

904^SB is an even integer; integer multiples of 1014 are even; the difference between two even numbers is even, hence cannot be 911 which is odd.

Alternative approach:

a^2 + 1/a^2 = 14.       (1)

From (1): a^2 = 14 - 1/a^2. Multiply through by a to get:   a^3 = 14a - 1/a.  (2)

From (1): 1/a^2 = 14 - a^2. Divide through by a to get:     1/a^3 = 14/a - a.  (3)

Add (2) and (3):   a^3 + 1/a^3 = 13(a + 1/a).    (4)

Now (a + 1/a)^2 = a^2 + 1/a^2 + 2 → 14 + 2 → 16 using (1)

Hence a + 1/a = 4. (5)

Put (5) into (4):    a^3 + 1/a^3 = 13*4 → 52

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"One alloy contains nickel and zinc in ratio 6:1, while another alloy contains nickel and zinc in ratio 20:21. If you mix the two alloys in ratio 23:15, what is the ratio of nickel to zinc in your new alloy?"

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