Number of solutions to this tricky equation?!

This is one of the hardest algebra problems I've seen on Twitter.

Problem from Twitter

I didn't mention the Lagrange polynomial in the video, but it is what inspired this problem

Math StackExchange post (which reminded me about the Lagrange polynomial)

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[MindYourDecisions]

You might be wondering how someone came up with such a strange equation in the first place! The answer has to do with a numerical analysis method called the Lagrange polynomial. The first three terms are exactly the formula for the Lagrange interpolating polynomial through the points (a, a^2), (b, b^2), (c, c^2). But the quadratic function x^2 fits these points too, and since 3 points determine a quadratic function, the Lagrange polynomial also simplifies to be x^2.




References



Lagrange polynomial


Math StackExchange post (which reminded me about the Lagrange polynomial)

[shubhayubasak]

Putting x=a, b, c satisfies....but since a quadratic equation has atmost 2 solutions we can say it is an identity...hence there are infinite solutions

[maksim-surov]

I multiplied the expression by (a-b)(a-c)(b-c), and after that collected all the terms with x^2, with x^1 and with x^0. All of them are zeros. The way is not much more difficult.

[tsuribachi]

I end up do it that hard way and expand the whole thing, which cause all the term that has a, b, and c to cancel each other out, leaving me with x^2 -x^2 =0

[TwelveOThirteen]

a≠b≠c is not a very rigorous notation of "a, b and c being distinct numbers" because it does not exclude the case a=c

[xiaoyang]

Well, the first 3 terms on the left-hand side (without x^2) forms the famous Lagrange interpolation polynomial. We can read off directly that it interpolates exactly at 3 different points a, b, c with values a^2, b^2, c^2 respectively. I.e., the resulting polynomial is just x^2.

[neutron]

It is called an "identity", an equation with infinitely many solutions

[sr.tarsaimsingh]

I know how to solve it before watching your video, this question is very frequent in our lectures of quadratic equation for JEE. Almost every math teacher bring this equation while teacher IDENTITY in quadratic equation.

[anoopmishra]

We can simply see that degree of equation is 2 but we found 3 solutions hence it is an identity and an identity have infinitely many solutions.

[pearlk]

Just put x = a, b, c all satisy the eqn so it's a 2nd degree eqn so there can't be 3 roots so this is an identify eqn and has infinite roots

[improving-homosapiens]

It is an entrance exam que JEE mains 16 March evening shift question. 😊😊😊😊😊😊😊😊😊.kind of question asked

[lakshaygupta]

we can see that x=a, b and c are the roots of the given equations, so it can't be quadratic or linear and hence it has infinite solutions.

[theevilmathematician]

However, you do not need this much work. Rearranging, we see a²((x-b)(x-c))/(a-b)(a-c) + b²((x-c)(x-a))/(b-c)(b-a) + c²((x-a)(x-b))/(c-a)(c-b)= x². Hence we get x²-x²=0 and there are infinite solutions. (I used this method btw).

[chetanrathaur]

Good Job JEE Aspirant! I know you solved the question within 3 seconds of seeing the thumbnail :)
A problem in every single JEE Module/book.

[michaelobrien]

My first instinct was to skip to the end! I saw that it was (at most) a quadratic polynomial and therefore should have 2 or fewer solutions. Immediately threw out D. But looking at it intuitively I thought it was likely infinite but was not sure how to go about solving ... and that expansion did not look like a good way to start my day! Since it was a 5-6 minute video, you weren't going to telescope that entire mess (telescoping videos are usually 4 minutes) or expand out (that would have been closer to 7 minutes) so a trick in-between must have existed! This was especially true since it took over a minute to say the problem! And thank you once again for showing another way to analyze a problem! Keeps me humble while letting me be challenged beyond the Math 9 I teach!

[shohamsen]

E. cause we already havbe 3 soln to a quadn(x=a, b, c) thus the quad is identically 0.

[Andreyy]

The answers hint a lot as they suggest the polynomial number of solutions does not depend on a, b, c. So without guessing for roots(not that is hard) we could replace a, b, c with trivial values and see how many solutions the polynomial has. In this case inf many. We wouldn't need the contradiction conclusion even. This shows how much information may be gained just by making a problem multi choice answers.

[vishalfgm]

hello i am currently a class 11 th student from india and with the knowledge of quadratic in depth i solved this easily as i have same kind of question in my study material and also we know that if there are more than 2 roots of a quadratic equation then that becomes an identity and has infinite many solutions

[BeattheCalculator]

Lagrange Polynomials never fail to impress! 😃

[alberteinstein]

This one stumped me

As do most of your vids