x + x*(2*x - x^2)^(1/2) + (2*x - x^2)^(1/2) - 1 solve , x ----> ? (Symbolic Evaluate)

[lvl107]

Hello Everyone.
From :

To :
       Symbolic Evaluate ? ( MC 15 )
Thanks in advance.
Regards.

[Werner_E]

Here you are, sure very informative and useful 🙂

BTW, the symbolics in Prime 10 doesn't even try to get an exact symbolic solution

[lvl107]

***.xmcd file, please.

[Werner_E]

Here you are

[lvl107]

Many, many thanks for ***.xmcd file, Werner. 😊

[Werner_E]

Here is a much simpler derivation.
In my first attempt i overlooked that the two root-expressions are identical.

So finally its only necessary to rearrange and square just once.

Interesting detail: While this worked (result too large..)

the expanded form forced muPad to float mode

[lvl107]

or another :

[Werner_E]

There is no need to use vectorization (while it does not do any harm) as x is not a vector.

As you had seen is muPad not able to simplify f(x) to 1 and you again ran into the nasty "...is too large ...".

But it seems that you can at least "prove" using muPad that the result is zero

The world isn't perfect ...

[LucMeekes]

Prime doesn't solve problems with order above 3 symbolically.

I guess it's time PTC put that wisdom in.

Luc

[LucMeekes]

Ah, no problem in Mathcad 11 (NOT Prime 11). Here are the four solutions:

Their numerical values are:

Success!
Luc

[lvl107]

My MC15 say :

 

and I don't understand the solution of 1.947 ( see my graph below )

Thanks any way, Luc.😊

[Werner_E]

---

and I don't understand the solution of 1.947 ( see my graph below )

😊

Maple in MC11 seems to be internally squaring the equation and so comes up with four solutions - similar to my modified appraoch here:

Re: x + x*(2*x - x^2)^(1/2) + (2*x - x^2)^... - PTC Community

Unfortunately, MC11 does not remove the invalid dummy solutions. See my reply to Luc below which shows where the 1.947 stems from.

By the way, Uncle Wolfram is also content with a numerical approximation only ...

.

[Werner_E]

Nice, but not surprising that Maple outbeats muPad.

Can you make the check if all four values really are solutions to the original equations?

I'm afraid that Maple is a little too generous with such equations and also gives dummy solutions that have to be dropped as solutions when checked.

It looks to me that 0.225 is the only solution - at least the only real valued one.

I may be wrong, but 1.947 doesn't look like a solution to me

In my approach when I was helping muPad by squaring the equation a few times and so getting rid of the squqre roots I ended up with eight solutions, the last 4 seem to correspond to the solutions you got with MC11

EDIT: I edited my approach and now come up with an equation of fourth order, yielding the very same results as MC11.

See Re: x + x*(2*x - x^2)^(1/2) + (2*x - x^2)^... - PTC Community

It can be clearly seen that 1.947 is not a solution of the original equation but rather of a similar equation with the signs of the root-expressions being changed:

[LucMeekes]

Interresting !

I haven't found a way to have Mathcad 11, when feeding back any of the symbolic solutions to the function f, to simplify the result to 0.

In all four cases the result is an expression several pages wide.

Luc

[Werner_E · Accepted solution]

This happens (more often) in MC15 as well that f(x) can't be simplified to 0 or a certain value c we know the expression should simplify to.

Also f(x)-c usually won't simplify to zero but most of the time the symbolics is able to simplify the Boolean expression f(x)=c to 1. Not sure why.