Can You Pass Harvard's Entrance Exam?

Рет қаралды 639,524

Ай бұрын

Entrance examination. If you're reading this ❤️.
What do you think about this problem?
Hello My Friend ! Welcome to my channel. I really appreciate it!
‪@higher_mathematics‬
#maths #math

Пікірлер: 351

@prithvichidri9634 Ай бұрын

For the guys who directly substituted x= -2🤙

@ernesthakey3396 Ай бұрын

Yep - math nerd with an MIT degree, just looking at it I figured X needs to be negative so that it turns into an addition, -1 doesn't work, -2 does, done. Of course for an exam question they probably require you to show your work, and you don't get credit for just saying "It is intuitively obvious!"

@vvTejedor Ай бұрын

Yes, but youcan say, that according to Vieta's formulas if there is a rational root, 12 can be divided to it.

@kutenor 27 күн бұрын

In my head in -2^3 seconds.

@arturkowalski5151 27 күн бұрын

5 seconds for me. And I have studied history, not mathematic...

@AasimSyedGPlus 24 күн бұрын

But that's not the only solution

@markjones5268 7 күн бұрын

The correct answer is: My parent’s just donated $5M to the school.

@aliahmed800 4 күн бұрын

😅

@divermike8943 22 күн бұрын

Oh ! He didn't calculate the Complex root! That's not the Harvard Way! The roots are X=-2, X= (3 +sqrt (15) i) / 2, X=(3-sqrt(15) i) /2

@ConceivedSorrow 22 күн бұрын

I solved it this way: x³-x² = 12 Extract common factor x²(1-x) = 12 12 can be written like 2²•3 (prime factorization) so: x²(1-x) = 2²•3 And comparte the factors: x² = 2² 1-x = 3 x= -2

@kingsircheng4540 20 күн бұрын

👍

@johnsmith1953x 19 күн бұрын

That's only one of the answers=FAIL.

@johannesvalks 18 күн бұрын

but it is mathematically incorrect. you miss 2 of the three solutions...

@lashamez 13 күн бұрын

Solution only works for integers, not real numbers

@thebig12conference73 9 күн бұрын

AND YOU FAILED! x^2 - x^3 = x^2 (1 - x) and 12 = 2^2 x 3 but the problem is that we choose x = 2, we don't have 1-2 = 3. No problem, just replace the 2 in -2 because 2^2 = (-2)^2. To recap, x^2 (1 - x) = (-2)^2 (1 - (-2)) , so -2 is a solution. Then, just factor by x - (-2 ) to get x^2 - 3x + 6 and check that there are no other solutions.

@tom-kz9pb Ай бұрын

Seeing -2 only took about 2 seconds. For Harvard, you should need to find the complex solution, as well.

@potrahsel4195 28 күн бұрын

Not sure about that.

@AllDogsAreGoodDogs 27 күн бұрын

Wolfram came up with x=-2 under "real" solutions. I agree with @potrahsel4195 .

@chrisclub3185 23 күн бұрын

Just divide the poly by x + 1 then use the quadratic formula on the resulting quadratic. If this is what’s required to get into Harvard, Harvard has fallen deeply

@chrisj5443 22 күн бұрын

Then it should be a harder problem. I'm dumb. It took me 5 seconds.

@suhumm 22 күн бұрын

@@chrisclub3185 by x+2, not by x+1 ))

@jbourdette 21 күн бұрын

The choice of 8 and 4 looks like magic. Be more formal using Gauss theorem about integer roots and soon get -2, the rest is easy, e.g. use Ruffini and Bhaskara.

@algorithminc.8850 Ай бұрын

I look forward to checking out your channel. Subscribed. Thanks. Cheers

@higher_mathematics Ай бұрын

Thank You Mister. Have a great day!

@robertfitzjohn4755 23 күн бұрын

I found x = -2 pretty quickly, but as it's a cubic equation it must have three roots. Surely to get into Harvard you have to do better than just say "oh, the other roots are complex, so I won't bother with them"? (Unless they specifically asked for only real roots, of course.) After a bit of work I managed to find the complex roots, and substitute them back into the original equation to prove they were correct. Then again, I did sit the Mathematics entrance exam for Natural Science at Oxford, some decades ago.

@TonalWorks 22 күн бұрын

Yeah, that was a wierd assumption indeed.

@lefterismagkoutas4430 20 күн бұрын

it's not that hard to find the complex solutions in the state that he left it, hell in most college classes they will leave it like that because it is almost implied.

@ericr2646 Ай бұрын

I worked out before I saw the youtube, on the right hand side of the screen, first of all X would have to be a negative number, so lets try X = -2, -2 squared is 4, -2 cubed is -8 take 4 - - 8 = 4+8 = 12, of course it is good to watch the youtube in case X is not so obvious.

@celestine5340 24 күн бұрын

Different approach: a bit trial and error gives x=-2 as a solution. So dividing the whole equation -x^3 + x^2 -12 = 0 by x + 2 gives -x^2 + 3x - 6 = 0. Which is quadratic, and thus solvable -> but has only imaginairy solutions. I think doing it this way is easier than the method used in the movie.

@ItsVideos 23 күн бұрын

Interesting. Thanks. Note that the video method also does a bit of "trial and error" when searching for powers of 2 that sum to 12.

@christianeb8494 17 күн бұрын

Exactement.

@barneynisbet6267 25 күн бұрын

-2 seems obvious. Then divide x3+x2-12, by x+2… then show that quadratic has no real roots….then find the complex roots. Or sketch the graph with intercepts and max/min and or points of inflexion….to show only one real solution… Seems like a beginner’s question to solving cubics.

@christianeb8494 17 күн бұрын

Tout simplement.

@The_Mega_AnalistSci 23 күн бұрын

I just do not know why is he doing that and not doing by Ruffini´s method which is x10 times easier and you get the result faster.

@lefterismagkoutas4430 20 күн бұрын

this is actually the quicker method if you know what a^3+b^3 is equal to, without needing to use any other method, he just does it very analytically.

@The_Mega_AnalistSci 20 күн бұрын

@@lefterismagkoutas4430 Have you even seen Ruffini’s method.

@Mirpurmad 3 күн бұрын

I loved it. brought back my school memories

@kumertd 25 күн бұрын

Gosh, what does higher mathematics mean in 2024..?

@cookiemonster-nk3xb 11 күн бұрын

Do you want a fresh one..mind your manners. If you want hard problems go to Presh Tell Walker at Mind Your Decisions 😂

@wisemang73 25 күн бұрын

Passes harvard math exam. Fails 1st grade writing

@gabbleratchet1890 22 күн бұрын

It's not really the entrance exam for Harvard. There is no such thing. They look at your grades and SAT/ACT scores.

@NainoLoL 22 күн бұрын

@@gabbleratchet1890 Or alternatively, your wallet 💰

@jayrussell3796 19 күн бұрын

@@mxm5783I actually commented that 2 days after your post...that scribble drove me crazy😅😅

@MariusBezuidenhout31337 22 күн бұрын

Order of operations (exponents before multiplication) would mean that -2 squared = -4. If x = (-2) it would be correct otherwise it would be x ≈ 2.68

@CuriousFocker 21 күн бұрын

I was going to post exactly the same thing. It's amazing the number of KZfaq maths channels that get things wrong.

@Gary.Holmes 21 күн бұрын

This is a really interesting point. I hadn't considered where the minus sign belongs in -2^2 but I do see the ambiguity, that -2^2 is not (-2)^2 . Is that what you mean? Thanks for making my brain itch! 😀

@jaycathedral9079 22 күн бұрын

Trig in high school and secured transactions in law school had one thing in common: Over the ensuing decades, I perhaps used each discipline twice. Algebra and calculus were a little more practical and I have yet to need to know the maximum area of a circle that can be inscribed in a square with a radius of X. Then again no one has asked me about the gram molecular mass of a molecular substance either. I took a lot of math and science and am glad I did. Not sure why. . .

@devastationofmankind3495 16 күн бұрын

So...for everyone thats not understinding the joke...yeah no me neither i quite literally have a grand total of ZERO idea what he's doing after the 4 step...but hey, neither do you...

@MandyFlame 22 күн бұрын

Factor directly as (x+2)(x^2-3x+6)=0 (By inspection x=-2 is a solution, then just work out the other bracket)! Then solve the second bracket using the quadratic formula to find the complex roots.

@TonalWorks 22 күн бұрын

Yeah, I did exactly the same just by inspection/ reasoning: (x+2)(x^2 + ? + 6) = x^3 - x^2 + 12. Btw: Why is he only solving for real x's? It's not obvious from how the equation is presented in this video. I hope the Harward problem was more clear regarding the set of solutions.

@santiagoferrari1973 22 күн бұрын

Much better than the long way in the video

@superserioes 3 күн бұрын

I was very bad at math at a normal school. Of course I wouldn’t have passed this test. But I’m positively surprised that not only did I guess -2 fairly quickly, but I also completely understood the proper solution here. For me that’s something, considering this is effing Harvard.

@loicgeeraerts 22 күн бұрын

I have another approach that does not require knowing the formula a^3 - b^3 and which allows you to find the solution -2 almost directly : x^2 - x^3 = x^2 (1 - x) and 12 = 2^2 x 3 but the problem is that we choose x = 2, we don't have 1-2 = 3. No problem, just replace the 2 in -2 because 2^2 = (-2)^2. To recap, x^2 (1 - x) = (-2)^2 (1 - (-2)) , so -2 is a solution. Then, just factor by x - (-2 ) to get x^2 - 3x + 6 and check that there are no other solutions.

@killianlomax8237 18 күн бұрын

Do a readable, well-written, straight X ... how hard can it be??

@bsharpmajorscale 21 күн бұрын

I actually tried a fractional answer at first, but then I remembered that odd powers change signs. Can't say as I'd have figured a complex answer except that Wolfram Alpha listed one when I checked my solution. Neat.

@bookert2373 27 күн бұрын

X² - X³ = X²(1-X) = 12. X² positive, so 1-X must also be positive, ∴ X < 1. For negative values, say X < -M, you have X² > M² and 1-X > 1+M, so 12 = X²(1-X) > M²(1+M). Plugging in values of M of 1, 2, 3, etc immediately shows M must be < 2, and that M = 2 (ie X=-2) is a solution. So reduce degree by dividing original by (X+2) to get quadratic and easy from there. This isn’t ’elegant’, but it only uses elementary algebra knowledge. It’s really all about finding that one solution X= -2 and reducing the poly degree, no matter how you find/guess that X=-2 is a solution,

@celestine5340 24 күн бұрын

yeah, that's exactly how I did it.

@bookert2373 24 күн бұрын

@@celestine5340 yes, one either guesses that -2 is a root or creates some simple inequalities as we did to find -2 is a root. After that, it’s easy. Knowing that there has to be at least one real root (since degree is odd) is a hint that looking for a real root is probably a good idea.

@tensor131 8 күн бұрын

Factorising gives x^2(1-x)=12, and associating the x^2 with the factor of 4 from 12 immediately yields the integer root. From there the quadratic quotient is easily found yielding the complex roots. Here in the UK that would be an easy question for a16 year old doing further maths A level.

@DanoshTech 20 күн бұрын

Bro for Harvard entry solving a cubic equation is very basic and i fell you should be required to find the complex solutions which are (3+-i√15)/2

@TheOldeCrowe 18 күн бұрын

1. kind of easy to see thar x² - x³ = 12 when x = - 2. 2. synthetic division then gives x³ - x² + 12 = (x + 2)(x² - 3x + 6) 3. the quadratic formula gives the two additional complex roots x = ½(3 ± sqrt(9 - 24) = ½(3 ± sqrt(15)i).

@christianeb8494 17 күн бұрын

Tout simplement.

@JustinTime-fr8fo 3 күн бұрын

imaginary math is so diverse and non discriminating it makes the word a better place.

@wilson2455 23 күн бұрын

certainly took the extra long way for something I did (in my head) in about 10 seconds !!

@stevenjohnson1143 Ай бұрын

Easy -2 squared is 4 minus -8 when subtracting a negative number flip both signs gets 12

@panszarlotka 24 күн бұрын

It can be done more simply. Just decompose the number 12 into factors. It will be 2 * 2 * 3. One of the factors will be the absolute value from X. Then just substitute all the factors into the equation and see if it is true for -2 or for -3.

@kjetilskotheim1712 24 күн бұрын

If we assume x is a whole number.

@jonr3198 23 күн бұрын

@@kjetilskotheim1712it's a relatively safe assumption if you're getting an exam with no calculator. That said it takes a few seconds to do this and if you got the answer you're done and if not then do it the complicated way

@saschaganser9671 18 күн бұрын

@@jonr3198 You must see that -2 is not the only solution directly. There must be complex solutions too. If you write -2, you didn`t even do half the job.

@gigapede 26 күн бұрын

It’s intuitive that x = -2, but if x would have been anything else I would have been lost on how to solve it, even though it looks so simple. Thank you for the tour of the solution, I need a lot of practice to become comfortable with those steps.

@stevenjeng1237 28 күн бұрын

It is quite simple! One root is -2, so (x+2)(x^2-3x+6)=0, then you can find 3 roots

@christianeb8494 17 күн бұрын

Et oui! Concours d'entrée pour Harvard? J'ai fait les Mines d'Albi-Carmaux, c'était tout de même un autre niveau.

@nobodynemoq 5 күн бұрын

come on, it took me a split second to figure out it's a small negative number and another second to ensure it's -2

@hotman718 21 күн бұрын

You can factor out one x set it equal to zero and solve it as a quadratic

@q-tuber7034 16 күн бұрын

There is a strange poetry in how the bizarre way of writing the letter “x” (completely missing the essence of the x) is echoed in the bizarre, overly complicated way of solving the problem

@surfer_guy942 22 күн бұрын

Use the rational zeros theorem and get x = -2, x = (3 +- sqrt(15) i ) / 2.

@KrystianMichalski 24 күн бұрын

x=(-2)

@andredcavalcante Ай бұрын

That's is a simple equation: what number the square is greater than the cubic form? Well, this number must be negative. Well... trying -1, square(-1) minus cubic(-1), 1 + 1 = 2, don't equals 12; trying -2, square(-2) equals 4 (promissing), cubic(-2)=-8, 4+8=12. I got it! The first real root is -2. I know may there are 2 other real ou complex roots, but I'm ok with these rapid and simple thoughts.

@nicolasbourbaki8896 18 күн бұрын

you should realize the answer x =-2 directly just by looking at it

@RaceSmokie 4 күн бұрын

Totally agree, it is obvious that x can not be 1 or 2 or 3 ,,, it must be in minus ,,, so -1 does not work, -2 ,,, voilà! What a waste of time with all these calculations, and who cares about complex roots, what is the use of it?

@wolfrogamer6116 3 күн бұрын

You see if you don't do this type of work you must realize that what your doing is not math but telling the answer or guessing the answer from your own experience

@RaceSmokie 3 күн бұрын

@@wolfrogamer6116 what's the point of writing a dissertation on something totally obvious at first glance. Is it that what they waste time on Harvard, to learn how to blablabla on worthless things or they learn how to make elephant from the fly. Where you will use all of these equations? I dont get it.

@wolfrogamer6116 3 күн бұрын

@@RaceSmokie haha so you think the study or math is useless or who study physics or maths are just a bunch of nerds Well in that case you are quit wrong because this "useless equation" are used everywhere. At least every where in the digital world. You see we talk to our computers even our mobile phone through these so called useless equation and if math had not existed how you would have gone through space or even figured out how old something is. I don't know about you but I am a computer science student and I have to create equation for my requirements and I use math to calculate almost everything. And I said that work is nessesary, I said it so because through these steps you can prove that you actually did it. You didn't just cheat off someone else. Now I do get your point that if you just want to know it by your self, you don't want to prove anything to anybody but even then you are still performing these steps and by experience you know the answer.

@RaceSmokie 3 күн бұрын

@@wolfrogamer6116 actually you are wrong bcz I dont think that math is useless. now, tell me where you have to use whole that process and what is use of it if you just can see the result without any calculation, what is a point to do all of this. Exactly where will be useful to do this equations. What to prove? Obvious things? Pure waste of time.

@danweingarten7674 14 күн бұрын

Far easier approach, factor both sides: (1) LHS: x²-x³ = x²•(1-x) (2) RHS: 12= 4•3 = 2²•3 From (1) & (2) we have x²=2² and (x-1)=3. Solving the second equation we get x=-2.

@tophat593 20 күн бұрын

Was following you right up to the cubic formula. Not sure how you concluded that.

@vict85 18 күн бұрын

x² - x³ is negative for x > -1, and the function is monotonically decreasing. Therefore, the function has a single negative real root. It is trivial to identify -2 as the real root. The complex root requires a polynomial division. Obviously, there exists a closed formula for the roots of any cubic polynomials, so...

@Noviscentur 3 күн бұрын

-2, элементарно. Даже не решал, сразу подставил ответ, он на поверхности.

@warrenbrown8670 18 күн бұрын

Did it in my head in under 10 seconds after I saw it. Interesting to see the process but some things you just see the answer.

@andytraiger4079 19 күн бұрын

Inspection reveals X must be a negative number for a real solution... and a very small negative number because X^3 blows up faster than x^2.

@user-so6eo6pg9v 12 күн бұрын

Even before I started to think, my intuition said me to check negatives. Then, it is a sum of square and third power. 4+8, so, the answer is -2. This is the solution if the problem is supposed for 12-years old children. For Oxford, you also need the complex ones

@ila9063 16 күн бұрын

There is no entrance examination for Harvard University.

@marcozoboli 15 күн бұрын

You can easily solve that equation noticing that x=-2 is a solution and then factorizing using the Ruffini's rule.

@rickwilson9508 8 күн бұрын

Solved it in my head. Super simple. Had to be negative number. -2 solved to 4 - -8 = 12. Sure there’s a sophisticated way but why when it’s easy

@frenzy1111 18 күн бұрын

There is in fact no question just an equation but assuming that the question is to solve for X, then X should be defined as an element of a number set. X€R, N or C or similar. Without this information we can't hope to answer any implied question.

@АлёшаИнкогнитов 24 күн бұрын

x² - x³=12 1-x=12/x² x=1-12/x², so X is rational. Also X

@bartermens8219 4 күн бұрын

You also draw 2 graphs x² ans -x³. And see where they are 12 apart. Sounds much easier than all the hokus pokus with x.

@alexkalish8288 Ай бұрын

well done, cool proof -

@thebig12conference73 9 күн бұрын

Engineering background from Berkely for those of you who think x=2 YOU CANNOT SOLVE THIS PROBLEM. I have another approach that does not require knowing the formula a^3 - b^3 and which allows you to find the solution -2 almost directly : x^2 - x^3 = x^2 (1 - x) and 12 = 2^2 x 3 but the problem is that we choose x = 2, we don't have 1-2 = 3. No problem, just replace the 2 in -2 because 2^2 = (-2)^2. To recap, x^2 (1 - x) = (-2)^2 (1 - (-2)) , so -2 is a solution. Then, just factor by x - (-2 ) to get x^2 - 3x + 6 and check that there are no other solutions.

@luciu1001 19 күн бұрын

Guess the x=-2 Factor the expresion using polynomes rules Solve for cases Way easier

@ethannguyen2754 10 күн бұрын

Rational roots theorem gives you x = -2 Divide the expression by x+2 to get x^2 - 3x + 6 and solve for the remaining roots with the quadratic formula. This is how most people who could solve this problem would do it. Your solution is interesting, but looking for those kinds of solutions would probably waste a lot of time on the exam. Especially if the more basic strategies for finding roots of polynomials work well enough.

@rrb79 19 күн бұрын

If square is greater than the cube then X is negative. Started with -2 for X and it worked. Next!!

20 күн бұрын

Yeah, the real root is quite elementary. Figuring it out took me just a couple of basic thoughts. The complex ones, weeell…

@Filimaxable 7 күн бұрын

х=-2, я решил в уме за полминуты простой логикой. квадрат числа минус куб получается положительное число. очевидно, что куб числа больше. уравнение имеет смысл, если х-отрицательный. квадрат так и так будет положительным, а куб - отрицательным. и получаем квадрат числа плюс куб числа равно 12, тут и думать нечего, что х=-2

@pmbpmb5416 9 күн бұрын

At a second glance -2 , if asked to show workings they would roll around the floor .

@eddievanbasten1751 5 күн бұрын

I worked it out in my head in less than five seconds.

@imdedicateddragon 22 күн бұрын

What? This is just a simple cubic equation

@satautenyo 9 күн бұрын

then the solution depends on FIGURING 12=8+4... in this case is also valid FIGURING that x=-2 is the first solution (trying integers by hand) and dividing the cubic equation by (x+2) which yields x^2+3x+6=0 and then the roots are 3/2 +- sqrt(15) i /2 as said by other mates... I don't think mine nor author's are valid and FORMAL solutions anyway...

@jerrypaquette5470 27 күн бұрын

It is pretty obvious that one solution is -2. Therefore X+ 2 must be a factor of X^2 - X^3 -12. Multiply -1 you get X^3 -%^2 + 12. Using division (X^3 _X^2 +12)/ (X+ 2) you get X^2 - #X + 6 X^2 - 3X + 6 = 0

@TipTop6996 9 күн бұрын

Hey I'm in! Guessed -2. Job done.

@darkboi5422 7 күн бұрын

For JEE it is taught that in 99% cases A root for a cubic equation is 0 1 2 -1 -2 and 0.5

@Kurrazade Күн бұрын

X^2(1-x) = 12 . The only multiplier of 12 that is a square is 4.

@tommymorton4939 Ай бұрын

1) X = -2. 2) Divide to (x+2), the result is square eqv. 3) Find two more complex roots.

@christianeb8494 17 күн бұрын

-2 est racine évidente, il n'y a plus qu'à faire une division polynomiale par (x+2) pour ramener cette situation à un polynôme du 2nd degré? Concours d'entrée pour Harvard? Pas besoin de monter une usine à gaz.

@ajhahn7765 16 күн бұрын

I remember doing this exact problem in my college algebra class. I didn't know I could get into Harvard with it LOL.

@BruceLee-io9by Ай бұрын

It's easier to break down the polynomial using Ruffini's method.

@alexlucky3338 Ай бұрын

Exactly my thoughts) but it's good to have an alternative method in mind as many people are "allergic" to polynomial division 😅

@EddyGilley Ай бұрын

I liked Raffini in #LionKing. When he held up Simba on Pride Rock before all the Pride Landers to see, it gave me chills. Circle of Life. #RaffinisRule 😉 Semper Fi

@drahcirnevarc9152 23 күн бұрын

I did a very mathematical philosophy PhD - it was on modality - 12 years ago at the age of 48. But I also quit maths at school 45 years ago after a respectable but not outstanding o-level. Suffice to say that I wouldn't pass Harvard's entrance exam! But then of course it's just patently obvious that 4 - -8 = 12.

@e-conelug5006 6 күн бұрын

I heard that today is enough to shout “ from the river to the see” and you are admitted.

@mrbigg151 20 күн бұрын

I can't tell you how many times this replayed 😂. The first time, I was looking at the wall for bullet holes

@jormungardwe 24 күн бұрын

I am very sorry for those that applied Cardano's formula

@FEPractice123 19 күн бұрын

by convertin x*x(1-x)=12 you will know (by inspection) that the solution must be a number less than 1. Thus, using x=-2 you have the solution in less than 30 seconds.

@jasonbarker1587 9 күн бұрын

I love Harvard, but the MIT in me found the answer in a quarter of the time.

@FreeSpeech181 4 күн бұрын

Why the heck would anyone need to know this if he or she is NOT majoring in math?

@mkill572 Күн бұрын

The real question here is what the uploader was high on that they didn’t get -2 right away by just putting it into the equation …

@TheFrewah Ай бұрын

Well, x^2(1-x) on one side and (-2)^2 - (-2)^3 = 4- (-8) = 12. So x=-2 , it has to be negative and you know 4+8 =12

@daniellemb1592 21 күн бұрын

I am very bad at maths, and had always been. I watched at x² - x³ = 12 and thought "x has to be negative, let's try with -2", so I calculated mentally and wrote 4+8=12. Seemed good ! So x= -2 Don't ask for the demonstration ! 😃

@RobertRoth-oj6zz Ай бұрын

After seeing problems and solutions like this, I don't even know how I got through algebra. I must have been shown a shorter way. I don't recall having to do problems this long.

@Ordinal_Yoda 27 күн бұрын

Yeah was cool. Cuz Mom taught me stuff like this in fourth grade. Best Mom ever! 😊❤

@electroaimant5402 7 күн бұрын

This method would not be allowed in France. You can never demonstrate smtg by assuming smtg else. In this instance, that 12 = 8+4.

@maartenverdouw4688 18 күн бұрын

Nice. Would be funny if you had chosen to rewrite 12 as 4 - -8

@billcook4768 Ай бұрын

Since when does Harvard have an entrance exam?

@redskytitan 19 күн бұрын

If Harvard had an entrance exam based on merit its Asian student population would approach 90%, at least 🤣

@16leviathan 17 күн бұрын

If the question ooks simple then the answer has something to do with negative or imaginary numbers.

@virag1132 23 күн бұрын

pro tip: harvard doesn't have an entrance exam lol

@baselinesweb Ай бұрын

Baffled everyone? You only have to substitute for x to get the real root, divide to get the quadratic and then use the quadratic equation to get the complex roots. Don't oversell.

@mohamedibrahim1836 9 күн бұрын

Mate, It takes 2 sec to solve this. you know it is going to be a negative because the cubic is larger in absolute than the square so x must be negative such that when cubed and subtracted it is a positive larger amount than the squared. 1 would undershoot, 3 will overshoot then it is 2 :D:D:D:D. The last step was not even thought of for me but just justifying why I immediately substituted with -2. Is that Harvard or 4th grade? But that is just one root out of the three ;) If they all overlap that need to be checked formally :D

@ThermalWorld_ 4 күн бұрын

This is called - Math zigzagging 😆

@nnamdiwilliams1498 21 күн бұрын

This was a little confusing to me. Where can I get resources to brush up on these things?

@raffaeleercole9630 18 күн бұрын

What about Ruffini’s method

@hotman718 21 күн бұрын

I would solve using properties of logarithms but many ways to solve this

@drmjruff 21 күн бұрын

It's -2. I guess if this is their difficult stuff the degree isn't worth that much.

@arijitmondal8605 9 күн бұрын

x= -2, (3+rt15i) /2, (3-rt15i) /2

@carlosceschini4104 22 күн бұрын

Es más facil resolver por teorema de taiz racional p/q y bajar de grado por Ruffini...

@taemagic 7 күн бұрын

-2 (just solve in my head)

@Name_Happrycorn 21 күн бұрын

Прошу вас объяснения по приложению этой информации на практике. Я имею в виду - применение формула разложения квадратного/ кубического многочлена на практике. В Гарвард я поступать не собираюсь. Спасибо!