What is it the second root?

[ValeryOchkov]

Sorry, I am not a Mathematician.

What is it the second root? See please the picture and Mathcad 15 sheet in attach.

Same in Maple

[AlanStevens]

Mathcad (MuPad) has solved it by first squaring both sides and then finding the two values of x that satisfy the resulting equation.  It hasn't then checked to see if both values of x are valid solutions of the original equation!  Reinforces the fact that you shouldn't automatically believe the results of computer calculations without checking!

Alan

[AlvaroDíaz]

Hi Allan

AlanStevens wrote:

Mathcad (MuPad) has solved it by first squaring both sides and then finding the two values of x that satisfy the resulting equation.  It hasn't then checked to see if both values of x are valid solutions of the original equation!  Reinforces the fact that you shouldn't automatically believe the results of computer calculations without checking!

Alan

Always! See for example: Re: No name curve

"Strange solutions" are those introduced by some algebraic manipulations in the process of isolating variables. For example, from

    x-1 = sqrt(3-3x),

you can get

    x^2-2x+1 = 3-3x,

then

  x^2+x-2=0

which gives x=-2 and x=1. But actually, x=-2 is a "strange solution" introduced by the algebraic manipulation, because don't hold the original equation, and must to be discarded. The only one solution for the original equations is x=1.

Best regards.

Alvaro.

[AlanStevens]

‌To be fair to Mupad, if you specify the values of a, b and c before the solve function, then Mupad only produces a single result.

ALan

[AlvaroDíaz]

Hi Allan.

You're right, seems like the solvers go back to the original equation for check only if the answer is numeric, but not when have some symbol. Probably because can't symplify enough the answer:

It could be interesting to check what happen if the symbolic answer have, for example, RootOf function, even all values are numerics.

Best regards.

Alvaro.

[tietjee]

x^2+x-2=0  has two solutions -2 and 1.  When you substitute back into the equation, make sure to include the minus sign is the square.

(-2)^2+(-2)+2=0

Cheers

[AlvaroDíaz]

Hi David.

Sure, but the issue appear when you need to use these equation for solve another, which don't have both of those solutions, like this example:

Here, -2 isn't a solution. It's a strange solution, introduced by the algebraic manipulations. Best regards.

Alvaro.

[ViktorKorobov · Accepted solution]

Can be nparameters a, b, c negative in the physical sense?

Viktor

[ValeryOchkov]

Viktor Korobov написал(а):

Can be nparameters a, b, c negative in the physical sense?

Viktor

See please fig. 5 here http://twt.mpei.ac.ru/ochkov/Optic.pdf

И большое мерси за уточнение.

Еще раз с Днем Варения!

ЗЫ Рад буду принять ваши замечания и дополнения по вышеотмеченной статье.

[ValeryOchkov]

One more question

[AlvaroDíaz]

Hi Valery.

In spanish literature we call those things as "soluciones extrañas" (strange solutions in english?).

We are introduced for that concept at very elemental levels. This example is for secondary school, isolating rational expressions: Ecuaciones racionales. Soluciones extrañas . Here you can see a definition of this concept, and elemental examples too. But can't found concrete references of this in english or russian. (There are some ones in french).

Best regards.

Alvaro.