Math Olympiad Problem | A Nice Algebra Challenge | You Should Know This Trick

In this video, I am presenting step-by-step solutions to this interesting and nice algebra problem. This is an algebra problem that you need to analyze polynomial equation, working on quadratic equation and quartic equation at the same time to come up with the answer. This video will walk you through how to analyze polynomial equation or algebraic equation, working with exponential operation to come up with the final answer. This is a good practice for math olympiad, too. Come check this video out and watch it until the end. More to come! Stay tuned!

💬 Did you enjoy this video? Tell me all about it in the comment section below!

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[danielpearson]

I substituted 1-a for b for b in a^2 + b^2 =2. Using the quadratic formula, one gets that b = 1 plus or minus the square root of three all divided by two. Then, with a lot of busy work one can solve for b squared using the one plus the square root of 3 over two which is, which is 2 plus the square root of three all over 2. Multiply by this itself to get b^4 and one gets 7 + the square root of three all over 4. Multiply this by itself to get b^8 and one gets 98 plus 56 times the square root of 3 all over 2^4. Its gets a little complicated after that, but then multiply b^8 times b^2 times b and one gets 989 + 571 times the square root of three all over 2^6 equals b^11. then solve for a using a= 1-the square root of three all over 2. using the same process one comes up with 989 - 571 times the square root of 3. all over 2^6. Woops, you add these two together and you get... 989 divided by 32. I must have made an error the first time that I did this. My bad.
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[justinnitoi]

I did it by solving for a and b directly, but this is more satisfying

[Nononow]

I first saw this problem on someone’s twitter and thought this is a very basic Algebra 1 systems of equations problem. After seeing that the question required that you find the value of the expression without solving either equation I appreciated how this is an application of substitution. I am left thinking that since the method is what matters why didn’t they create a system with a clean integer value when you find the value of the expression.

The values of a and b are not terribly interesting (although the larger of these two irrational numbers is approximated nicely by the ratio of the boiling and freezing point of water in Kelvins).


Although some people may instantly recognize the solution for the first two equations I think that the system below would be much more satisfying to engage with.

If a+b=1
and a^2+b^2=3
then what is a^11+b^11

[kummer]

I solved it using the binomial theorem. Takes more steps but gets the job done.

[MrGLA-zsxt]

I used substitution but got the same answer. Cool video

[Min-cvnt]

this is just beautiful and fantastic prof.

[danielpearson]

I got 857 plus 76 times the square root of 3 all divided by 32. This comes out to 988.64 divided by 32. I did this a different way. Why is my answer so close but different?

[domedebali]

Elegant explanation indeed. Nice work professor