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It is a lot easier tonfactorize 1680, you get 2,2,2,2,3,5,7 to get 4 numbers in sequense an you know 5 and 7 are two of the, if you try 4,5,6,7, you willhave one 2 left over so you get 5,6,7,8, 5x6x7x8=1680 so x+6=8 x must be 2 done
I did it pretty much the same way. Took me 20 seconds in my head, maybe less. Since it is a factorial, we know that x+ 2 is a positive integer, and we know that 1680 must be the product of four consecutive integers. It's obvious that combinations like 2-3-4-5 and 3-4-5-6 are too low, so we quickly multiply 4-5-6-7 (still too low) 5-6-7-8 (just right) . Since we know that x+6 (the highest number in the sequence) is 8, x must be 2. I like your additional bit of logic, stipulating that 5 and 7 must be two of the factors., therefore the only possibilities are 4-5-6-7 and 5-6-7-8. I didn't think of that. I think that would have slowed me down, because I can multiply faster than I factor, but well played!
You're looking for 4 sequential numbers that multiply to 1680. So you know the 4 answers have to be close to the 4th root of 1680, or about 6.4. Doesn't take long to guess 2 smaller numbers and 2 bigger numbers, for a run of 5/6/7/8. So x+6=8, and x=2.
The purpose of such videos is to demonstrate a systematic method of solving such problems. Otherwise he could have easily said, "after doing my calculations, the only REAL answer is 2".
Take the fourth root of 1680. It is 6.4. Then divide 1680 by 6*7*8 and you'll get 5. That x+6 = 8. He did it the most complicated way. If you calculate the discriminant then why not use the formula.
Une fois écrit (X+3)(X+4)(X+5)(X+6) = 1680 Peut-on écrire 1680 en produit de 4 nombres consécutifs ? 1680 = 2^4x3x5x7 = 5x(2x3)x7x(2x2x2) = 5x6x7x8 Donc X = 2. Bingo !
@@yung-kanglee4681 except that factorials are only defined for non negative integers or if you get rather fancy in your math you can extend it to non negative numbers as a whole but that gets kinda funky and beyond the scope here i think. either way you's ended up with (-5)!/(-9)! plugging that in which is not defined in either the numerator or denominator and as such is not actually a solution to the problem
La décomposition en nombre premier de 1680 c'est 16*3*5*7, comme c'est le produit de 4 entiers successifs, entre 5 et 7, il y a 6 si on divise 1680 par 5*6*7 qui vaut 210 on obtient 8 donc 1680 = 5*6*7*8
Magic! I didn't do it the "mathy" way but figured that the difference in factorials is 4 contiguous numbers (6-2). Factored 1680 to 2,4,5,6,7 which is 5 numbers, but aha!, 2*4=8, so it's 5,6,7,8, so x+6=8, and so x=2. Less aggravation.
You know that the fraction will give you an x^4 plus some other stuff. So, then you realize that if x is >/ 7, x^4 > 1680 so the answer has to be less than 7. Then just substitute values for x and find very quickly x = 2.
Нет смысла решать два квадратных уравнения. Арифметический квадратный корень не может быть отрицательным. Т.е. надо была сразу заменить u и решать одно уравнение
It's a (provocatively?) complex solution. ;) As it has already been written here, the obvious idea is to factorize the right part. Given that x+2 is natural, we immediately come to the only solution x=2.
Didnt watch the video, but solved it like this (the way with least possible calculations): 1) It's obvious that we are looking for 4 numbers in sequence with a product of 1680. 2) Obviously, 10*11*12*13 is already too big, because it is larger than 10000. 3) Also we can easily see, that the sequence does not contain 9, because we can divide 1680 by 3 only one time. So it can only be 5*6*7*8 or lower. 4) But on the other hand, the sequence MUST contain 8, because 1680 can be divided by 2 more than 3 times. So it can only be 5*6*7*8. OR you just guess it :)
You write 1680 = 8.7.6.5 and equate it to (x+6)(x+5)(x+4)(x+3). The solution x = 2 would be simpler. But if there were other solutions they would be more difficult to find. Thanks for the algebra.
