solving x^4+1=0

We will solve x^4+1=0 by factoring. This is one of the most interesting quartic equations because it will give us all the fourth roots of -1. Enjoy!

0:00 WolframAlpha couldn't factor x^4+1 without complex numbers
0:26 the first way
4:25 Learn on Brilliant
5:29 the second way

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[blackpenredpen]

Here’s a harder video: x^4+x^3+x^2+x+1=0

[MuPrimeMath]

The complex way is very cool, I didn't think to try that!

[G-k]

Because of you i became addicted to math i even became able to get very high scores on exams, Thank you so much from Morocco.

[Sanmboazzz]

Honestly, ever since I started following this site, I'm learning great Math tricks!

[acn]

Me who remembers the video where he calculated the square root of i

[ashishpradhan]

"Maths is not about solving for X
It's also figuring whY"

[toaster]

I discovered this first method when I noticed that x^6 - 1 can be factored as a difference of cubes or a difference of squares first, leading to the polynomial x⁴ + x² + 1 in one result and the factored version of it in the other result. I never would've realized it could be factored otherwise.

[ItachiUchiha-ipen]

I just come to see his smooth switching between the pens🤩

[strawberryfluff]

Your videos are very informative, amazing, and fun to watch!

[batuozer]

What about just
x^4=-1
x^2= +-sqrt(-1)= +-i
X = +-sqrt(+-i)

[MrDerpinati]

for complex roots, using de moivres theorem is probably a good way, sadly however ive literally already forgotten it even though i did it about a week ago

[Amoeby]

Saying things like "everyone knows i^2 is equal to -1, but you need to know that -1 is the same as i ^ 2" with a beard like that, you look like a math sage

[rogierownage]

Alternative way at 2:16:
Move (sqrt(2)x)^2 to the other side and take square root on both sides. Don't forget to take +- on the right because you are taking the square root.
x^2+1 = +- sqrt(2)x
Then move the right side to the left side.
x^2 +- sqrt(2)x + 1 = 0
Which gives you the same result that one of those two must be zero.

[raefgormley]

Where did the -1 come from in the second line? I was expecting (x^2)^2 + 2x^2 + (1)^2 - 2x^2

[Barchueetadonai]

I feel like Method 2 would have been much easier to simplify by replacing i with e^(i*pi/2) and - i with e^(i*3pi/2). Then the square root of them is the same as raising them to the 1/2 power, which would rotate them to (with the + - for each) e^(i*pi/4) and e^(i*3pi/4). From there, their values in cartesian via some trig would be +-(cos(pi/4), i*sin(pi/4)) and +-(cos(3pi/4), i*sin(3pi/4)). which line up exactly with Method 1.

[AnatoArchives]

Me and my friend in school tried to solve this equation too! It took us days for us to solve it and had the same solution!

Now knowing these techniques in algebra made it easier! Thanks bprp!

[vardaandua]

You could have done it directly writing -1 on the other side and using Euler form ....and using the concept of n th root of unity

[uno-x-oun]

I love how you still love to hold your poke ball while talking, I did that when I was in 2nd grade.

[stephenbeck]

Have you done a video on DeMoivre’s Theorem? Quick way to get to this result once you understand powers of complex numbers as repeated multiplication using cisθ or e^iθ form.

[davinderSingh-zrhu]

Can u find common area under curves x|x| +y|y|=4, x|x|-y|y=4, y=|x| please...