The FASTEST Way to Solve Cubic Equations (Math Hack)

Ajouté : 2026-05-12

[00:00:00]This equation is not as simple as you think.

[00:00:04]But believe me, I am going to show you every step I am taking to find all the possible solutions of X.

[00:00:11]Pay close attention to this. This equation is very very interesting.

[00:00:15]We have X to the power of 3 divided by 8 is equal to 1.

[00:00:19]Okay? So, this is a cubic equation. That means we we must have three solutions.

[00:00:25]This is very interesting. So, this implies that X cubed will be equal to what? 8 * 1 is what? Is 8.

[00:00:34]So, we have transformed it in this form.

[00:00:37]Now, we are going to move 8 to the left-hand side. We shall have X cubed.

[00:00:43]Now, this 8 can be written as what? -2 to the power of what? 3.

[00:00:48]Because 2 to the power of 3 is 8. All this is equal to 0.

[00:00:53]Now, let's consider this algebraic identity.

[00:00:56]If you have A squared or A to the power of 3 minus B to the power of 3, this is the same thing as saying A minus B multiplied by A squared plus AB plus B squared.

[00:01:13]Okay?

[00:01:15]So, let's use this identity to solve this.

[00:01:19]Okay? So, where our X is equal to A, then our B is equal to that's 2 to the power of 3.

[00:01:31]That's um sorry, 2.

[00:01:33]Which is 2.

[00:01:34]So, put in this into this and see what we shall have.

[00:01:39]Our A is X, so we shall have X minus B is what? 2. So, by 2 multiplied by A is X, so we have X squared plus um AB, which is what?

[00:01:55]Um X * 2, which is 2x.

[00:01:59]Plus b squared is 2. That is 2 squared, which is what? 4.

[00:02:05]All this is equal to zero.

[00:02:07]So, we we we shall obtain the first solution because x - 2 will be equal to zero, which means x is equal to 2.

[00:02:16]That's for x1.

[00:02:19]This is interesting, right?

[00:02:21]Now, the second solutions or the other solutions is by solving this quadratic equation.

[00:02:27]So, we shall have x squared plus 2x plus 4 is equal to zero.

[00:02:35]We cannot solve this through a factorization because we cannot um factorize this.

[00:02:40]But, we can solve this by using quadratic formula.

[00:02:44]So, our quadratic formula said that x is equal to minus b plus or minus square root of b squared minus 4ac divided by 2a.

[00:02:56]Right?

[00:02:57]Now, in here we shall obtain Look at very well. We shall obtain our uh a a to be equal to 1, then b to be equal to 2, then our c is equal to 4.

[00:03:15]Putting these parameters in this in this uh quadratic formula, we shall have that our x will be equal to Our b here is 2. So, we shall have minus 2 plus or minus b squared. Our b is 2.

[00:03:31]That means we shall have the square root of 4 minus 4 * um our a is 1 * c. Our c is 4. So, 4 * 1 * 4 is 16. So, we shall have 16 here divided by what? By 2.

[00:03:51]So, our a is 1. So, we shall have two times one, which is two.

[00:03:56]Right. So, look at this.

[00:03:58]So, X will be equal to -2 plus or minus Now, 4 -16 is -12. So, we shall have the square root of -12 divided by two.

[00:04:15]Right?

[00:04:16]Now, look at this.

[00:04:18]This is a complex solution because of this -12.

[00:04:21]Let me show you how to express this.

[00:04:25]So, this become X will be equal to -2 plus or minus This is the same thing as saying 2i the root of three.

[00:04:40]Okay?

[00:04:41]This minus the root of -12 is 2i root of three.

[00:04:48]Right? Now, all this is divided by what?

[00:04:51]By two.

[00:04:53]Now, we can factorize this out as our X will be equal to um two into -1 plus or minus I multiplied by the root of three divided by two. Okay? So, that this two will neutralize this two.

[00:05:12]Then, we shall have our X to be equal to -1 plus or minus I root of three.

[00:05:22]So, our X1 will be equal to -1 plus I root of three or our second solution will be -1 minus I root of three. So, you can call this your X2. You can call this um your X3.

[00:05:46]So, these are the solutions to this amazing equation. I told you earlier that it is not that simple because we are going to obtain all the solutions of x, which is a cubic equation. So, our first solution is x is equal to 2, our second solution is -1 + um -1 + i root of 3 or -1 - i root of 3.

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