Why so much calculations ,by factor/remainder theorem we get immediate answer x=-2 then by synthetic division we get another quadratic factor which can be easily solved and get two complex roots
This is very good approach - in advanced mathematics I would use the Rational Roots Method to find x = -2, then divide the cubic by x+2 to find the quadratic,
First, we will divide the expression by minus to get -X^2+X^3=-12 and X^3-X^2+12=0, (X+2)(X^2-3X+6)=0. One of the roots equals -2, and the other two are determined from the expression X^2-3X+6=0 by using the quadratic formula.
Any number squared is a positive, a number cubed can be either polarity. By inspection, x must be negative. Let's try -2. (-2)^2 - (-2)^3 = 12 4 - (-8) = 12. It fits.
Thanks for the video ! Great thinking 💜 I did it slightly differently : after I realised that -2 is a root (by just looking at the polynomial), which means that (x³ - x² + 12) is divisible by (x + 2), I just performed the division and found (x² - 3x + 6), and then found the two complex solutions in the same way. Not sure I would have thought of using the identities for (a³ + b³) and (a² - b²), but I’ll try to remember that 😃
A lot of people down here thinking the answer is just '-2'. I think there are clues in the title with the phrase 'Most Fail' indicating that the teacher is not looking for just the REAL solution. If you gave just -2 as your answer you would FAIL to get into Harvard and end up going to a community college instead. :)There are 3 solutions.
Depends on what the question asked was - was it "give all possible answers" or was it "find a solution for x". "Most fail" was the caption on the video.
Interesting and well presented until the last few seconds when a promotion for another of JJ's videos obscures the portion of the whiteboard where she is concluding her work.
Factorising the cubic elements and by using the difference of two squares she's broken it down into one linear multiplied by one quadratic equation equals zero.In order to get zero one of them must be zero. She's pretty good. Next up completing the square to prove the quadratic formula- easy sums!Good video.
Thanks for the content you are really good at math. I understood how you resolved it even though I didn't do math since a long time, but it would be great for others if you explain more slowly, that would make you a great mathematician and a great teacher.
X=-2 by inspection. So x+2 is a factor (factor theorem). Divide the expression =0 by x+2. This will yield 2 complex roots. Basic differentiation will allow curve sketching to confirm only 1 real root. Really no need for loads of algebra. Especially if it’s a Harvard entrance problem, which I seriously doubt.
I used to think like that too. You may not use equations like this in real life however, you will become sharper in solving problems in other areas of life.
It takes about three seconds to see that x = -2 solves this. It is a cubic equation, of course, so it will have three solutions. The other two will be complex, though.
Intuitively the answer is x = -2 The procedure might seem long but if done mentally will give you the solution within seconds. x² - x³ is always negative if the value of x is more than 1. It's always positive if the value of x is less than 1. Now for anything between 0 and 1, we'll have x² - x³ also between 0 and 1. That would mean the value of x must be a negative number. If we write x as -k, then we get k² + k³ = 12, where k is a positive real number. Now all we have to do is take two or three guesses. Taking k as 1 will give us less than 12. Taking k as 2 gives us exactly 12. Anything more than that will be more than 12. This means k can have only one value, that is 2. This gives us x = -2. By our reasoning, there can't be any other real value of x. Now a real cubic equation will either have 3 real solutions or 1 real and a pair of complex solutions. This means the other roots are complex conjugates.
Thank you, that was fun. The best part of it is if it was on a college board test where you did not need to show Work, the guess and check method would be very simple because you would know that the imaginary part would not work as an answer. So just take the -2 and that would be the only one that does work. Always loved doing college algebra
I did this in about 30 seconds in my head. Harvard must be a cakewalk. The trick was that the answer had to MINUS due to any POSITIVE answer would not work since x cubed is > x squared, so the answer would always be < 0. -2 came instantly in my head as a guess and it solves the problem. Seriously? JK about Harvard - total respect for that institution.
From the comments, it is clear that at least about 2/3 of them come from people who never heard about imaginary numbers, neither understand that there must be three solutions to the equation. An interesting aspect is the psychology behind these, mostly cocksure, responses - these individuals don't know what they don't know.
Start with a negative number, as sqare is positive and cube is negative. So both terms are positive and add up. Thus, it is x sqare plus x cube is 12. So x is 2.
X must be negative because whilst the square of a negative number is positive, the cube is negative. Futhermore, the cube is subtracted from the square to equal 12. 12 is a small number, so x must be smaller. In this case it's 2. 2 -(-8) = 12. It takes longer to type than to analyse and calculate mentally.
The is one Real root that is -2, and two Complex roots that need to be calculated. So the answer x=-2 is wrong. And generally speaking a polynomial equation in x³ will always have 3 roots. You don't need to go to Harvard to know that.
You can do it in your head. At a glance, you can figure out that an abvious answer is x = -2. If you divide (-x^3 + x^2 + 0x - 12) with (x+2), you get (-x^2 + 3x - 6). Problem solved in 2 steps.
Is this Harvard or Hardward? Answer of -2 in a couple of seconds. You know it has to be negative. It is not -1, and -2 fits in nicely. Cubic equation and hence 3 solutions for X. You figure X=-2. Factorize and solve tge quadratic to get the imaginary numbers. This will be a 1 min video.
I guess harvard wants to see you regurgitate and apply basic math formulas to determine you atleast know the basics. Im also guessing there are many of these questions on the exam with very little time to aee if you can handle stress not being able to finish. This question called for no real thinking,,its like riding a bike...
..a is used to represent x....like a placeholder. It is not part of the solution to the problem. If my explanation still leaves you confused, ignore a completely and focus on how she solves the problem with x and the numbers, and you will understand it.
Some of you at the comment section are talking trash 🗑️🗑️, will you please make your own vedio with all the qualities that you are talking about so that we can appreciate!!!note:The vedio should be of the same question for us to make comparisons!!😂😂
Straight forward; good explanation. You explained everything you did, and you backed it up with the formulas that you would use.
She is demonstrating methodology. It becomes more relevant when solving more complex problems which cannot be solved at a glance.
Thank you
How brilliant you are ! Your experience & explanation is very very best.
The other girls are dancing on TikTok and jumping from hotels to hotel
Thank you for taking the time to make this video. Much appreciated.
Harvard doesn't have an "entrance exam".
With all the X's everywhere, we still don't have 24hr electricity anywhere.
Thank you so much for explaining it so well and patiently.
Regards 🙏
U are welcome
Great presentation
Why so much calculations ,by factor/remainder theorem we get immediate answer x=-2
then by synthetic division we get another quadratic factor which can be easily solved and get two complex roots
This is very good approach - in advanced mathematics I would use the Rational Roots Method to find x = -2, then divide the cubic by x+2 to find the quadratic,
First, we will divide the expression by minus to get
-X^2+X^3=-12 and X^3-X^2+12=0,
(X+2)(X^2-3X+6)=0. One of the roots equals -2, and the other two are determined from the expression X^2-3X+6=0 by using the quadratic formula.
x = - 2, (3 + - √- 15) /2
This method is shorter
Any number squared is a positive, a number cubed can be either polarity.
By inspection, x must be negative.
Let's try -2.
(-2)^2 - (-2)^3 = 12
4 - (-8) = 12. It fits.
Thanks for the video ! Great thinking 💜 I did it slightly differently : after I realised that -2 is a root (by just looking at the polynomial), which means that (x³ - x² + 12) is divisible by (x + 2), I just performed the division and found (x² - 3x + 6), and then found the two complex solutions in the same way. Not sure I would have thought of using the identities for (a³ + b³) and (a² - b²), but I’ll try to remember that 😃
👍👍
A lot of people down here thinking the answer is just '-2'. I think there are clues in the title with the phrase 'Most Fail' indicating that the teacher is not looking for just the REAL solution. If you gave just -2 as your answer you would FAIL to get into Harvard and end up going to a community college instead. :)There are 3 solutions.
Depends on what the question asked was - was it "give all possible answers" or was it "find a solution for x". "Most fail" was the caption on the video.
Who is a lot of people. It's only you who thinks so.
Great! I was wondering how to solve it, thanks for a clear explanation.
you are a smart lady!!!
Interesting and well presented until the last few seconds when a promotion for another of JJ's videos obscures the portion of the whiteboard where she is concluding her work.
Oh! So sorry about that.
Right😊
@rajamnaidu1962Look to the far left. It is there!
@@JJONLINEMATHSCLASSchannel:
It's the same all over this platform, don't worry.
@@JJONLINEMATHSCLASSchannelyou know your stuff.
Factorising the cubic elements and by using the difference of two squares she's broken it down into one linear multiplied by one quadratic equation equals zero.In order to get zero one of them must be zero. She's pretty good. Next up completing the square to prove the quadratic formula- easy sums!Good video.
God bless you sister.
Further Math...knew all of this and had an A. Never used it in real life 😂
Thanks for the content you are really good at math. I understood how you resolved it even though I didn't do math since a long time, but it would be great for others if you explain more slowly, that would make you a great mathematician and a great teacher.
one root is -2. not sure about the two others, but i am sure you will explain in video
X=-2 by inspection. So x+2 is a factor (factor theorem). Divide the expression =0 by x+2. This will yield 2 complex roots. Basic differentiation will allow curve sketching to confirm only 1 real root.
Really no need for loads of algebra. Especially if it’s a Harvard entrance problem, which I seriously doubt.
Very well done. Thank you.
It's not difficult in learning this stuff, but what's it use in a the real world situation?
I used to think like that too. You may not use equations like this in real life however, you will become sharper in solving problems in other areas of life.
Airplanes, cars, computers, telephone etc etc would NOT exist without math
It takes about three seconds to see that x = -2 solves this. It is a cubic equation, of course, so it will have three solutions. The other two will be complex, though.
12 = (-2)^2 -(-2^3)
X^2-X^3 = (-2^2) - (-2^3)
X = -2
Only one root?
Intuitively the answer is x = -2
The procedure might seem long but if done mentally will give you the solution within seconds.
x² - x³ is always negative if the value of x is more than 1. It's always positive if the value of x is less than 1.
Now for anything between 0 and 1, we'll have x² - x³ also between 0 and 1.
That would mean the value of x must be a negative number. If we write x as -k, then we get k² + k³ = 12, where k is a positive real number.
Now all we have to do is take two or three guesses.
Taking k as 1 will give us less than 12.
Taking k as 2 gives us exactly 12.
Anything more than that will be more than 12.
This means k can have only one value, that is 2.
This gives us x = -2.
By our reasoning, there can't be any other real value of x.
Now a real cubic equation will either have 3 real solutions or 1 real and a pair of complex solutions.
This means the other roots are complex conjugates.
x=-2 or x = (3 +/- sqrt(15)i)/2 where i^2=-1. Hint: use synthetic division.
the correct answer for x is - 2
There are two, well three answers, 2 is the observe one if you have any brains.
good stuff, I enjoyed it
wow, I had taken time without going through this Mathematics, thank you, I see all the steps we used to do while in High school
U are welcome
excellent
Thank you, that was fun. The best part of it is if it was on a college board test where you did not need to show Work, the guess and check method would be very simple because you would know that the imaginary part would not work as an answer. So just take the -2 and that would be the only one that does work. Always loved doing college algebra
-2 in a few seconds
Excellent thank you...👍
U are welcome
Good job my sister and make me proud.
I did this in about 30 seconds in my head. Harvard must be a cakewalk. The trick was that the answer had to MINUS due to any POSITIVE answer would not work since x cubed is > x squared, so the answer would always be < 0. -2 came instantly in my head as a guess and it solves the problem. Seriously? JK about Harvard - total respect for that institution.
One value of x is -2. So x+2 is a factor. By dividing I would find the other quadratic factor. Will factorize ......
nice. It has been over 45 years since I did that. I now remember.
I used to be very good in mathematics but didn't expect l was gonna see i ever again.
Thanks you so much for your incredible help in calculus
That's not calculus
This is painful
For instances where the answer is not as simple as -2, this more elaborate approach is probably applicable :-)
As long as you must have "i" in the "solution" it is not a real solution
Because x^3 is bigger than x^2, x must be negative. The factors of 12 are 1, 2, 3, 4, 6 ,12 and their negatives. Try -1, -2 ,,,
-2works.
From the comments, it is clear that at least about 2/3 of them come from people who never heard about imaginary numbers, neither understand that there must be three solutions to the equation. An interesting aspect is the psychology behind these, mostly cocksure, responses - these individuals don't know what they don't know.
In the competetive exams, this type of questions are to be solved within 30 to 40 seconds. So, after guess and try, simple answer is -2. That's all.
Well this one is easy. The first observation is that x
If it is I expect you are supposed to solve it by inspection, which shows a far better understanding than using a rote method.
You would have to know all the theories,Etc.,to solve this type of problem.
X2(1_x)=4×3.
1_x=3.
_x=3-1=2.
X=-2
thank you
U are welcome
Wow, I forgot about the math entirely.
My dear guess the first root.
Divide to find the quotient
Use quadratic formula to solve the quotient.
What the hell I just witnessed? One glance is enough to see that x=-2.
😂
x = -2 satisfies the equation
Only the real part.
Why do I need this to study accounting at Harvard...or anywhere else???
BRILLIANT!!!
if x
Please don't use the quadratic equation. As thinking people, we should complete the square. Please teach us how to do that!
I have videos on completing the square method. Please check
Do you privet tutoring??
Start with a negative number, as sqare is positive and cube is negative. So both terms are positive and add up.
Thus, it is x sqare plus x cube is 12. So x is 2.
(1-x)x^2=3*4,
x^2=4, x=±2,
1-x=3. x=-2.
Is it x=-2 ?
X must be negative because whilst the square of a negative number is positive, the cube is negative.
Futhermore, the cube is subtracted from the square to equal 12.
12 is a small number, so x must be smaller. In this case it's 2.
2 -(-8) = 12.
It takes longer to type than to analyse and calculate mentally.
x= -2 in my head < 5 seconds
Why hide the answer by adverts to another video?
That is a bother she had address, otherwise good video
very good
It s obvious that -2 is a solution of this équation. So let factorize by X+2....
👍👍
x^2-x^3=12 x=-2 x=(3±Sqrt[15])/2i=1.5±0.5Sqrt[15]i final answer
X must be negative
And answer is -2 just by inspection.
It is a 3rd degree polynomial, so there are 3 roots. -2 is just one of them.
The key is knowing that the cube can be either polarity, which makes -2 obvious....
x^2-x^3=12
x^2(1-x)=12=2×6=2^2×3==>x=-2
Very interesting.
Especially the "i"
👍👍
Brilliant. Obviously used to the board, algebra and the marker. You belong to Harvard.
Wow it's interesting
Thanks
If x = -2 then we have 4 --8 = 4 plus 8 equals 12. At least that's,a solution.
👍👍
Can u go further by using surds methods to go beyond the answer to get discreet values?
The is one Real root that is -2, and two Complex roots that need to be calculated. So the answer x=-2 is wrong. And generally speaking a polynomial equation in x³ will always have 3 roots. You don't need to go to Harvard to know that.
Nonsense. Where did she say you have to go to Harvard to know that x^3 won't have 3 roots?
Everybody is somebody's X you just need to find Y
I'm 20 seconds in and I can already intuit that the answer is -2.
I find it hard to believe that JJ has subscribers. James Joyce however makes perfect sense.
x^3 - x + 12 = 0
so
x^3 + 2x^2 - 2x^2 - 4x +3x + 12 = 0
so (x+2)(x^2 - 2x +3)=0
so x = -2
or x = [2 +- sqrt(4-12)]/2
= 1 +- isqrt2
That's what I also found
You can do it in your head. At a glance, you can figure out that an abvious answer is x = -2. If you divide (-x^3 + x^2 + 0x - 12) with (x+2), you get (-x^2 + 3x - 6). Problem solved in 2 steps.
That’s what I did.
It’s -2 is it not?
Only way a square minus a cube can be positive is if x is a negative number.
Nice one. But this is why I ran away from Maths lol. 🤦
🤣
Lol. Puts the pop-ups over the final answers.
factorise. 3x4=12 therefore x = -2
x=-2,the other roots are imaginary.
Life's too short.
Really too short!!! I'm none the better for her explanation 😢😢 jah jah!!
Pls where are you based. My mathematics is a bit rusty now!
What is the i exactly please used in the quadratic formula, please
'i' is a complex number, it's the square root of negative one
❤❤ you are doing well , my colleague , A mathematician .
Thank you so much
I resolved it 1min in my mind.the result is X={-2}@@JJONLINEMATHSCLASSchannel
Thank you so much.
U are welcome
Is this Harvard or Hardward?
Answer of -2 in a couple of seconds.
You know it has to be negative.
It is not -1, and -2 fits in nicely.
Cubic equation and hence 3 solutions for X. You figure X=-2. Factorize and solve tge quadratic to get the imaginary numbers. This will be a 1 min video.
Came to the same conclusion, 6th grade level math and that's harvard standard? 😂😂
I guess harvard wants to see you regurgitate and apply basic math formulas to determine you atleast know the basics. Im also guessing there are many of these questions on the exam with very little time to aee if you can handle stress not being able to finish. This question called for no real thinking,,its like riding a bike...
Hee Haw
Where did a come from?
..a is used to represent x....like a placeholder. It is not part of the solution to the problem.
If my explanation still leaves you confused, ignore a completely and focus on how she solves the problem with x and the numbers, and you will understand it.
Mental arithmetic! By initially splitting 12 into 2 x 2 + 2 x 2 x 2 you have arrived at the answer, the rest is beating about the bush!
But that ignores complex numbers
Some of you at the comment section are talking trash 🗑️🗑️, will you please make your own vedio with all the qualities that you are talking about so that we can appreciate!!!note:The vedio should be of the same question for us to make comparisons!!😂😂
thank you