An algebra problem from Finland

In this video, I solved an interesting algebra Olympiad problem from Finland 2008.
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[@punditgi]

Prime Newtons has solved the problem of how to teach math! 😊

[@SpiroGirah]

Always enthusiastic. I really admire your joy and happiness. I wish you all the best in life!

[@BartBuzz]

I think you solved it the way the problem was meant to be solved. You always prove that math is fun...if one enjoys solving puzzles!

[@saba6601]

Prime Newtons-you are really a marvelous guy who makes MATHS BECOME ALIVE. Regards Dr.Sabapathy (Mathematician, Singapore 🇸🇬).

[@PrimeNewtons]

Thank you!

[@tao7289]

I’m from Singapore!

[@duckyoutube6318]

My fav math channel. Always a good puzzle in the mornings.

[@syamantagogoi]

I really admire you for the way you solve the equation. You have made Math quite interesting through your excellent articulation on each and every problem. Please keep it up.

[@rantainenkoivu]

Nice video as always! Greetings from Finland :)

[@AriaA.-ub5jt]

I love the soothing tone in your voice, makes math so much more enjoyable

[@PrimeNewtons]

Thank you.

[@mahinnazu5455]

Thank u very much Sir. Mahin from Bangladesh.

[@aguyontheinternet8436]

I did it by squaring, pulled the graph up on desmos to see if I missed anything, and then realized I clearly missed something when a solid 12 units of the line was just y=2. I then found the perfect squares and absolute values. 6:08 I mean even if case 3 had a solution it would be invalid as there is no real number greater than 4 yet less than 2

[@olivierbarthe1531]

I love the way you explain things with joy and relaxation!

[@jidraffgithendu7546]

Prime Newton's, you make mathematics look so easy

[@rajesh29rangan]

Temptation was to square both sides. But your solution is elegant

[@Komolification]

Wonderful.. You are a great lecturer and motivator..

[@saarike]

Simply Great!!! It was a joy.

[@Moj94]

I tried going a bit further and replaced one of those sqrt(x+1) with sqrt(x+2) to see whether I could solve it. I regret my decision.

[@doug95124]

A slight variation of the problem produces very surprising results. If you change it to sqrt(17+x-8*sqrt(x+1))+sqrt(5+x+4*sqrt(x+1))=6, then the solution seems to be -1

[@dirklutz2818]

Indeed! Very strange.

[@FadkinsDiet]

When I have an equation with multiple absolute values I graph it (by hand) to see where the sharp turns. Then it's easy to verify that you have all solutions.

[@NothingMaster]

I Finished this one, before I start it. ❤

[@SidneiMV]

t = √(x + 1) was a nice trick

[@ankiy5689]

Yeahh, It's an form of substitution strat used in Integrals

[@prabhatrexkira398]

Love from India ❤❤❤

[@jamesharmon4994]

I loved how you solved this, but I would like if you had gone back and plugged the x values into the original problem to verify they were not extraneous solutions. They're not, but this is an important step.

[@TheFrewah]

I remember this guy in university that should calculate appropriate dimensions for something that had length and diameter, maybe a cable. He confused the two such that the calculated diameter was much larger than the length and forgot to check if the answer was reasonable.

[@aguyontheinternet8436]

it is clearly impossible for any of the solutions shown to be extraneous. At no point did he make any action (such as squaring) that would basically combine equations together, which is how extraneous solutions form. The reason there are often extraneous solutions when attempting to solve these square root problem, is because in the process of solving them, you often square both sides and combine a different equation with your own. For an example, squaring the equation √(x+1)=√(x-1) combines your original equation, √(x+1)=√(2x-1), with a different equation, √(x+1)=-√(2x-1), and creates the equation x+1=2x-1 which has the solutions of both, although sometimes the extra equation added does not have any real solutions, like here one equation has a real solution and the other doesn't, though this isn't always the case.

[@jamesharmon4994]

@aguyontheinternet8436 I agree that many times it is extremely unlikely. However, if you are not under a time constraint, it never hurts to verify. I grew up with the following mantra: Trust but verify. I've found this to be handy for more than just nuclear disarmament. Extraneous solutions can creep in due such things as "arithmetic errors" such as swapping a minus sign for a plus sign. Verification would root these out.

[@user-rm9bb2xb4f]

bro thats like ez math for grade 9 in Vietnam

[@holyshit922]

If you solve it by squaring both sides gives the same result but one of them leads to complex radicals Your method is good, even better than squaring both sides

[@keinKlarname]

For me, it should be checked if the solutions really do belong to the given case (which means: they really are solutions).

[@PugganBacklund]

Feels like you should have confirmed that t=6 is valid for the base function (case 1 is valid for t >= 4), and that t=0 is valid (case 2 is valid for t

[@happyhippo4664]

Are the shirts only for Patreon members? Do you have hippo-sized 3XL?

[@PrimeNewtons]

Shirts are for everyone. Thank you. See description.

[@aguyontheinternet8436]

we have made a happy hippo today

[@nixheb]

First 30 seconds of the video, t'was litterally me.... :D

[@dirklutz2818]

A new trick every day! Love it!

[@elkincampos3804]

you should consider that t=sqrt(x+1)>=0 and t-2>t-4. t-20 is wrong.

[@YoutuberClips95]

Dont know why you used the letter "t" but fair enough

[@aguyontheinternet8436]

easier to spot the perfect square. otherwise you'd need to spot that 17+x-8√(x+1) = x+1-8√(x+1)+16 = (√(x+1)-4)^2 which is not incredibly easy to do without the substitution

[@PrimeNewtons]

What letter do you recommend?

[@YoutuberClips95]

@@PrimeNewtons Probably A or Y cuz they're more iconic 😂

[@nothingbutmathproofs7150]

Sorry but I don't buy your logic. You said in case 1 that both t-4 and t-2 are both positive. The problem is that when you concluded that t=6 you needed to check that when t=6 you had both t-4 and t-2 positive (or 0). Same comment for case 2.

[@niloneto1608]

Lmao is it that hard to verify when t=6, both t-2 and t-4 are non-negative?

[@aguyontheinternet8436]

Does it really require conscious effort to determine 6-4 is greater than zero?