Solve example on various calculators
12-10-2023, 03:19 AM (This post was last modified: 12-10-2023 03:35 AM by C.Ret.)
Post: #7
 C.Ret Member Posts: 276 Joined: Dec 2013
RE: Solve example on various calculators
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.

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 Messages In This Thread Solve example on various calculators - lrdheat - 12-09-2023, 02:38 AM RE: Solve example on various calculators - Albert Chan - 12-09-2023, 01:19 PM RE: Solve example on various calculators - lrdheat - 12-09-2023, 03:04 PM RE: Solve example on various calculators - C.Ret - 12-09-2023, 06:19 PM RE: Solve example on various calculators - lrdheat - 12-09-2023, 07:24 PM RE: Solve example on various calculators - Marco Polo - 12-09-2023, 10:08 PM RE: Solve example on various calculators - C.Ret - 12-10-2023 03:19 AM RE: Solve example on various calculators - Gene222 - 12-12-2023, 04:21 PM RE: Solve example on various calculators - lrdheat - 12-10-2023, 04:32 AM RE: Solve example on various calculators - carey - 12-10-2023, 09:27 AM RE: Solve example on various calculators - Albert Chan - 12-10-2023, 04:05 PM RE: Solve example on various calculators - lrdheat - 12-10-2023, 02:28 PM

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