What is it the second root?

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.

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.

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.