Solving a cubic equation with a parameter

This video is about solving a cubic equation and finding all solutions.

Polynomial expressions,difference of squares,algebra,algebraic equations,SyberMath,algebraic manipulations,equations,substitution,Challenging Math Problems,Non-routine Math Problems,polynomial equations,an algebra challenge,algebraic identities,non-standard methods,symmetry,math,maths,mathematics,Polynomial equations,an algebraic challenge,cubic equations,How to factor a cubic using an unusual method

[neuralwarp]

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

[lucanina]

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

[a.raghavan]

Isn’t there an error when we take sqrt 3 into cube root, you can’t multiply by 3.

[seegeeaye]

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.

[michaelempeigne]

sqrt ( 3 ) * cbrt ( 7 ) is not equal to cbrt ( 21 )

[elias]

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?

[kaslircribs]

You have a very nice way of solving problems. I like your method sir. Thank you.

[kamaljain]

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.

[neuralwarp]

What happened to the sqrt 3? Did you distribute it into the cube root?

[michaelempeigne]

you cannot combine a square root and cube root like you did.

[hp.basketball]

How do you solve a depressed cubic equation with unknown parametres example 2x^3 +(2k-1)x -4h=0