You: okay, right, good, so, well, what we're gonna do is, hopefully you can already see, right, no, actually, I'll tell you that later, awesome, ... Me: JUST GET ON WITH IT
A smart Italian mathematician: *discovers the cubic formula so that people could solve cubic equations quickly and easily* People: Screw it, let's solve it in a smart way: The guy who discovered the cubic formula: Am I a joke to you?
One way I discovered it has only one solution is to take the derivative of the expression and the a parameter gets away finding is zero at x=1/2 and always positive otherwise, that means that the cubic expression is of the form (cx+b)^3 +(a -b^3) with c and b to be calculated. but you solution is more elegant and clever
The cubic equation has one real solution and two complex solutions, or three real solutions. We rewirte the equation as 2X^2(2x -3) = a - 3x, the both sides as graphs, they intersect at only one point.
You're right! Thanks for the heads up! That escaped my attention. The correct way to combine would be to use the 6th root and write sqrt(3) times cube root of (2a-1) as 6th root 27(2a-1)^2
when you got the first solution, it has cuberoot of a real number. there are three cuberoots of a real number. the other two you get by multiplying the real cuberoot with w and w^2, where w is a cuberoot of unity. so right there you had all 3 solutions.
Isn’t there an error when we take sqrt 3 into cube root, you can’t multiply by 3.
That's an error!
😂
You: okay, right, good, so, well, what we're gonna do is, hopefully you can already see, right, no, actually, I'll tell you that later, awesome, ...
Me: JUST GET ON WITH IT
ok
What happened to the sqrt 3? Did you distribute it into the cube root?
I think I did!
😂
A smart Italian mathematician: *discovers the cubic formula so that people could solve cubic equations quickly and easily*
People: Screw it, let's solve it in a smart way:
The guy who discovered the cubic formula: Am I a joke to you?
😁
One way I discovered it has only one solution is to take the derivative of the expression and the a parameter gets away finding is zero at x=1/2 and always positive otherwise, that means that the cubic expression is of the form (cx+b)^3 +(a -b^3) with c and b to be calculated. but you solution is more elegant and clever
The cubic equation has one real solution and two complex solutions, or three real solutions.
We rewirte the equation as 2X^2(2x -3) = a - 3x, the both sides as graphs, they intersect at only one point.
sqrt ( 3 ) * cbrt ( 7 ) is not equal to cbrt ( 21 )
You're right! Thanks for the heads up! That escaped my attention. The correct way to combine would be to use the 6th root and write sqrt(3) times cube root of (2a-1) as 6th root 27(2a-1)^2
You have a very nice way of solving problems. I like your method sir. Thank you.
You’re welcome! Thanks for the kind words! 💖
when you got the first solution, it has cuberoot of a real number. there are three cuberoots of a real number. the other two you get by multiplying the real cuberoot with w and w^2, where w is a cuberoot of unity. so right there you had all 3 solutions.
Nice!
you cannot combine a square root and cube root like you did.
That's correct!
How do you solve a depressed cubic equation with unknown parametres example 2x^3 +(2k-1)x -4h=0