Solve example on various calculators

12102023, 03:19 AM
(This post was last modified: 12102023 03:35 AM by C.Ret.)
Post: #7




RE: Solve example on various calculators
I have only had my Ti30X 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 \( f(x)=(x+1)\times\sqrt{x^22x1}\). 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 \( p(x)=x^22x1\) 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 \(g(x)=(x+1)(x ^22x1)\) whose trace is in red on the figure. The polynomial \(g(x)=x^3x^23x1\) is of the third degree and its three roots can be obtained with the POLYSOLV module. These are the same roots as the line \(x+1\) and the parabola \(x²2x1\) 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 (NUMSOLV) 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  12092023, 02:38 AM
RE: Solve example on various calculators  Albert Chan  12092023, 01:19 PM
RE: Solve example on various calculators  lrdheat  12092023, 03:04 PM
RE: Solve example on various calculators  C.Ret  12092023, 06:19 PM
RE: Solve example on various calculators  lrdheat  12092023, 07:24 PM
RE: Solve example on various calculators  Marco Polo  12092023, 10:08 PM
RE: Solve example on various calculators  C.Ret  12102023 03:19 AM
RE: Solve example on various calculators  Gene222  12122023, 04:21 PM
RE: Solve example on various calculators  lrdheat  12102023, 04:32 AM
RE: Solve example on various calculators  carey  12102023, 09:27 AM
RE: Solve example on various calculators  Albert Chan  12102023, 04:05 PM
RE: Solve example on various calculators  lrdheat  12102023, 02:28 PM

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