I follow all of this, but have a problem with what the original question is actually asking.

With the question that started this thread: "x^3 - x = 20, find x to 1dp", isn't is asking "what value of x makes the value of x^3 - x closest to 20?".

Here is an example: "x^3-2(x^2)+2x = 5 has a solution between 2 and 3. Find x to 1dp."

x = 2.1 gives the cubic equal to 4.641

x = 2.2 gives 5.368

x = 2.15 gives 4.993375, which suggests that, because 2.15 is too small 2.2 is the better answer.

Yet, if you look at how far away we are from the 5 that we are aiming for, x = 2.1 has an error of 0.359, while x = 2.2 has an error of 0.368. Surely this means that 2.1 is a better value for x because the value of the cubic is closer to 5 ?

Or not?

[Probability of a typo in this post? Fairly close to 1 I should think ...]

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