Solving x^4=1

How to solve the 4th degree equation x^4=1? How many solutions does x^4=1 have? How do we solve x^4=1 by factoring? Can we just take the square root both sides for the equation x^4=1?

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#math #algebra #mathbasics

[NuzranRahat]

Its really easy when you watch the "x^3=8" video were he shows you all the solutions to that equation. But this one showed me that there are more ways you can solve these equations. Thank you ❤❤❤

[timeonly]

2:59 OMG!! The FIRST TIME I've ever seen BPRP flub a marker color change!! It'll go onto my list of "What were you doing when such-&-such event happened?". 😂😂❤

[marcogiai-coletti]

Your second method explains the concept really well.

[rotemlv]

Another way is to make a visit to trig land and use De Moivre's tools. Probably the most annoying way to write on paper but it is my favorite, as it helps you see the geometric properties of the roots of a complex number.

[Nikioko]

Quick way using Euler's formula:
x⁴ = 1
In the complex plane, the four solutions form a regular tetragon (= square), inscribed by a circle with the radius ⁴√1 and the centre at the origin:
x = ⁴√1 ⋅ [cos(n ⋅ 360°/4) + i ⋅ sin(n ⋅ 360°/4)]
n = 1→4

x₁ = 1 ⋅ [cos(90°) + i ⋅ sin(90°)] = 0 + i ⋅ 1 = i
x₂ = 1 ⋅ [cos(180°) + i ⋅ sin(180°)] = −1 + i ⋅ 0 = −1
x₃ = 1 ⋅ [cos(270°) + i ⋅ sin(270°)] = 0 + i ⋅ (−1) = −i
x₄ = 1 ⋅ [cos(360°) + i ⋅ sin(360°)] = 1 + i ⋅ 0 = 1


Long way using algebra:
x⁴ − 1 = 0
Apply third binomial formula:
(x² + 1) ⋅ (x² − 1) = 0
Apply the identity i² = −1 on the first bracket:
(x² − i²) ⋅ (x² − 1) = 0
Apply the third binomial formula twice again to get the fully factored form:
(x + i) ⋅ (x − i) ⋅ (x + 1) ⋅ (x − 1) = 0
According to the rule of the zero product, the whole product is zero if one of the factors is zero. So we get:
x₁ = −i ∨ x₂ = i ∨ x₃ = −1 ∨ x₄ = 1

𝕃ₓ = {−1, 1, −i, i}

[obinator]

if you want to troll someone solve this using the general quartic equation

[tank_man]

I revised complex numbers a few weeks ago, but you completely caught me off-guard

[emconstrucao.]

Another way of solving it would be to set y=x², and then y²=1, y=±1, x²=±1, x=±√±1. Finally x=1, -1, i and -i

[Chordrider]

He is fast but still easy to follow. Virtuoso use of the whiteboard. Just enough talking. I don't need to learn any math but I watch anyway. Good fun!

[DonTheRealMan]

Mathematicians: *stumped by negative square roots*
Rafael Bombelli: Just calls √(-1) "i" and moves on...

I love how based mathematicians are🤣

[zerid]

Another way (that works well with values other than 4) is to notice that |x^4|=|x|^4=1
So |x|=1.
Therefore we can write x in the form x=e^(iy)
x^4=1 means e^(4iy)=1
Which means 4y is a multiple of 2π so y is either 0, π/2, π or 3π/2 modulo 2π which is exactly 1, i, -1 and -i

[okaro]

I did it geometrically what angles multiplied by 4 give 2pi*n: 0, pi/2, pi, 3pi/2 so you get 1, i, -1, and -i.

[magnusmalmborn]

The polar method generalizes to all powers, so maybe do a video with it?

[siener]

I think the simplest, most general and most useful way to solve this is to write 1 in it's polar form and then take the natural logarithm on both sides, and the rest is easy.

Then you can replace the "4" with a "n" to get the general form for the roots of unity. The general form is also pretty easy to visualize: There are always n solutions, evenly spaced on the unit circle.

[rdspam]

Roots of unity - e^((2*pi*k)/n)i. So e^(1/2, 1, 3/2, 2)*pi*i, or i, -1, -i, 1

[thej]

You are so smart! Also you are an amazing teacher and presenter. It humbles me how much math I do not understand. This channel is always making me think.

[BeeDaWorker]

Show the polar method pls because y not
Also I'm about to learn it soon so it'd be nice to see that stuff applied in a simpler question like this

[Yootlander]

He really is crazy with the marker swap

[kururhai]

unrelated to video but I'm glad I found your channel currently our class completed calculus 1 and is teaching calculus 2, your explanation of questions really helped me and cleared up previous queries about some integrals and all doubts

[Lightning]

A more complex way of doing this: r^(1/n)=r(cis(Θ)). The 4 theta values would be 0, 90, 180 and 270. Input and solve