Solve example on various calculators

[C.Ret]

I have only had my Ti-30X PRO for less than ten days and I discovered this astonishing calculator which is neither graphic, nor programmable nor CAS. But very well made, the only criticism is its screen which seems a little small to me compared to those to which I have become accustomed.

A CAS machine allows us to understand what is happening here and provides a way to obtain the three solutions using a polynomial resolution module.



In blue the plot of the function \( f(x)=(x+1)\times\sqrt{x^2-2x-1}\).
This function is the product of two polynomials, the second of which is under a square root: \( f(x)=(x+1)\times\sqrt{p(x)}\)
The pale yellow area is the region where the function f(x) is not defined, because the polynomial \( p(x)=x^2-2x-1\) under the radical gives a negative value \(p(x)<0\).

As \( √(p(x))=0 \) is possible if and only if \( p(x)=0 \), the roots of \(f(x)\) are the same as the roots of the polynomial \(g(x)=(x+1)(x ^2-2x-1)\) whose trace is in red on the figure.
The polynomial \(g(x)=x^3-x^2-3x-1\) is of the third degree and its three roots can be obtained with the POLY-SOLV module.
These are the same roots as the line \(x+1\) and the parabola \(x²-2x-1\) whose traces are in green on the figure.That is to say, as Albert Chan already told us, the exact values of the roots are -1, 1-√2 and 1+√2.

Like Irdhead when I use the numeric solver module (NUM-SOLV) I only get the roots -.41421 and 2.41421 by entering the exact guesses 1±√2 respectively. Other guesses inevitably lead to a No Sign Change Error.

I observe the same thing on my NUMWORKS which also can only find the three roots using the degree 3 polynomial trick and cannot numerically find the roots 1±√2 with its general equation solver when the radical is used.


All this seems completely natural to me, I do not know of any numerical root resolution algorithm which does not need the function to be defined, continuous or differentiable on either side of each root.