OpenTube

Can You Find ALL Solutions? – Polynomial Equations

Math Queen

18,037 views

In this math video I (Susanne) explain how to solve a fourth-degree equation with four solutions, including two real solutions and two complex solutions. We use the imaginary unit i to write and solve the complex solutions. Mathematics explained.

00:00 Intro – Solve the equation

4:34 See you later!

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20 Comments

@wastrelway3226 20 hours ago

Well, dang it, I was going to expand the parentheses. But as soon as Suzanne said, "Let's take the square roots of both sides", I said, "Oh snap!"

@jlr43100 21 hours ago (edited)

Love her. Memory is coming back. Thank you. Always struggled with roots.

@Nostromo2144 11 hours ago (edited)

This was 3 times easier than I thought it would be! ;)

@gkotsetube 9 hours ago (edited)

Complex numbers really were the first YouTube attraction, centuries before YouTube. People who don't believe math is useful for anything in real life, get excited with complex solutions of equations.

@backwashjoe7864 4 hours ago

Thanks Susanne for another nice video. This inspired me to read about the Fundamental Theorem of Algebra for the first time in a long time. Its history is more complex than I’d imagined! I’m also curious now about graphs that include the complex plane. I’ve seen some good visualizations of that in the past, so need to find and review those. Thanks for expanding my mind this morning! You are a wonderful teacher. 😊

@alexiiitb 20 hours ago

Thank You, I like sooooo much the way You explain, You make it look so easy and pleasant at the same time ❣

@arhimed5603 15 hours ago

Hello from Odesa, Ukraine. Thank you for the fun math.

@HarryC-Smith 15 hours ago

That √(-11) really threw me at first, I was expecting √(-9)! I had to work it out in my head in both directions to understand why that wouldn't work!

@stur448 21 hours ago

Ok. If I expand this out, I’d get a x^4 term, so I will start assuming 4 solutions. Working backward, I can make 100 from 10^2 and -10^2, thus x^2 should be 9 or -11. Finally, x could be 3, -3, and if imaginary roots are allowed, root11•i and -root11•i.

@jim2376 19 hours ago (edited)

+/- 3 are the obvious real answers by inspection.
Expand for x^4 + 2x^2 + 1 = 100
u = x^2
u^2 + 2u + 1 = 100
Solve the quadratic u^2 + 2u - 99 = 0. u = -11 or 9
Back substituting 9 brings us back to our obvious real answers and for - 11 the complex answers of +/- sqrt(11) i

@Teacher_juliorondino 18 hours ago

Amazing 👏👏👏✨️

@SmokingSevens 11 hours ago

I got +3 and -3 off the top of my head. Interested to watch and see if there’s more.

@noobilatornoobie9114 13 hours ago

7:31 Funny . I am in France , and usually the i IS in front of any number (z= a + i.b)
(advantage: immediate identification of the imaginary part )

@mrolson1890 9 hours ago

Before I started the video, I looked at the "thumbnail" for ten seconds and came up with 3, which in the end turned up to be x1.

@jan-willemreens9010 21 hours ago

... (X^2 + 1)^2 = 10^2 [ A^2 = B^2 => A = B v A = - B ] => (1) X^2 + 1 = 10 <=> X^2 = 9 <=> X1 = - 3 v X2 = 3 ....

... (2) X^2 + 1 = - 10 <=> X^2 = - 11 [ i^2 = - 1 ] => X^2 = 11 * i^2 <=> X3 = - Sqr(11) * i v X4 = Sqr(11) * i ....

S = { - 3 , 3 , - Sqr(11) * i , Sqr(11) * i } ... ✌️👍

@rasheednajjab4076 7 hours ago

Thank you for your detailed well explained educational solution
Actually this brings us to a sort of conclusion that X jn the starting equation is raised to 2 Twice that it is X to the power 4
And that means we have to find 4 Values for X in this example 2 values are real while the others are imaginary or complex numbers as you described them, Thanks again

@uni-byte 19 hours ago (edited)

3, -3. If you can't see these immediately, you really need to spend time here. The imaginary roots are a little harder. Keep coming back, she will teach you.

@Moonraker6646 1 hour ago

I have to watch it again

@hemrajue3434 19 hours ago (edited)

What is fourth root of -1 ?
(x^2+1)^2=100
x^4+2x^2+1=100
x^4+2x^2-99=0
Let x^2=a
a^2+2a-99=0
a^2+11a-9a-99=0
a(a+11)-9(a+11)=0
(a-9)((a+11)=0
a-9=0 or a+11=0
a=9 --> x^2=9, x=+or -3
a+11=0 ---> a=-11 , x^2=-11, x=+or -i√11.