Algebra. Can solve the equation?, :-) Опции

[Williams]

Daughter. 8 class. Brought home from school. Said I bet you can't solve it.

1/x^2 -1/(x+1)^2 =1

or



Don't use Wolfram / Mathematica ... Test yourself :-)

The equation is from a series of "math tricks" turned out to be quite famous. I even watched a couple of videos about its solution. And remembered the reciprocal equations! Had a wonderful two hours. But I lost the bet! :-)

Added (solutions):

» Click to show Spoiler - click again to hide... «

https://www.wolframalpha.com/input?i=1%2Fx%5E2+-1%2F%28x%2B1%29%5E2+%3D1

https://www.youtube.com/watch?v=EwwcxAyHceU

[marianconstantin...]

Dear Williams,

IMO such tricks destroy the confidence in learning math, because a legitimate question would be "what if I am not able to see it?"...

"Your" equation leads obviously to a fourth degree polynomial equation. There are techniques to factorize the 4th degree polynomial form into two polynomial quadratic.

One of them is to force the 4th polynomial to get a difference of squares. This is not a math "trick", it is a healthy way to learn math, IMO.
So I suggest to take a look to the Michael Rozenberg's post in the following link. The hint is to introduce a constant k that should be calculated by a condition "delta=0" of a second quadratic form as to be a square. It is true that the equation giving "k" is a cubic equation, but in regular cases k is an integer and can be easily seen. For your case k=1.
The link:
https://math.stackexchange.com/questions/3282632/how-to-factor-a-fourth-degree-polynomial

Best regards.

[Williams]

marianconstantinescu73, totally agree. But remember, this equation is from a high school algebra course. Where as I remember this kind of thing (factorization) hasn't been studied yet.

[marianconstantin...]

Well, I'm not familiar with math curricula of 8th grade high school in Russia. In my country factorization by square differences for polynomial terms is accessible for 8th grade high school (more likely with difference of squares of polynomial 1st degree). For sure such way is accessible to 9th degree.

Anyway, I get older and despite I have a PhD in a field full of math, I'm still not able to understand the purpose of such problems that give rather barriers to learning than real knowledge. My opinion, of course.