Need help with few questions

Algebra

Need help with few questions

Postby sunny » Sun Jul 28, 2019 1:03 am


Hi I need help in solving the following equation. This question is Picked up from subject BCS12 Page 6. I have solved it on paper but I am not able to understand some logic. Please see the paper for further details.

The answer given in Book is :

[tex]W^{2}[/tex]+W=-1

because

[tex]W^{2}[/tex]+W+1=0

For my question please refer to the attachment. I have tried solving but not clear with the logic
Attachments
Unit 1 Question.jpg
Unit 1 Question.jpg (67.16 KiB) Viewed 130 times
sunny
 
Posts: 5
Joined: Sun Jul 28, 2019 12:42 am
Reputation: 0

Re: Need help with few questions

Postby sunny » Sun Jul 28, 2019 1:08 am

Unable to post 2 images in same post so posting img from book which contains answer
Attachments
book question and answer unit 1.png
book question and answer unit 1.png (6.26 KiB) Viewed 128 times

sunny
 
Posts: 5
Joined: Sun Jul 28, 2019 12:42 am
Reputation: 0

Re: Need help with few questions

Postby Baltuilhe » Sun Jul 28, 2019 1:55 am

Good night!

[tex]\omega[/tex] is the imaginary cube root of unity.

All the numbers that are solution to:
[tex]x^3-1=0[/tex]

So:
[tex]x^3-1=(x-1)(x^2+x+1)=0[/tex]

Solving:
[tex]x=1[/tex]

This is one solution. There are another 2:
[tex]x^2+x+1=0\\
\Delta=(1)^2-4(1)(1)=-3\\
x=\dfrac{-1\pm\sqrt{-3}}{2}\\
x=\dfrac{-1\pm i\sqrt{3}}{2}\\
x'=\dfrac{-1}{2}+i\dfrac{\sqrt{3}}{2}\\
x''=\dfrac{-1}{2}-i\dfrac{\sqrt{3}}{2}[/tex]

If [tex]\omega=\dfrac{-1}{2}+i\dfrac{\sqrt{3}}{2}[/tex], so
[tex]\omega^2=\left(\dfrac{-1}{2}+i\dfrac{\sqrt{3}}{2}\right)^2=\left(\dfrac{-1}{2}\right)^2+2\cdot\dfrac{-1}{2}\cdot i\dfrac{\sqrt{3}}{2}+\left(i\dfrac{\sqrt{3}}{2}\right)^2\\
\omega^2=\dfrac{1}{4}-i\dfrac{\sqrt{3}}{2}-\dfrac{3}{4}=\dfrac{-1}{2}-i\dfrac{\sqrt{3}}{2}[/tex]

So, the solutions are:
1, [tex]\omega[/tex] and [tex]\omega^2[/tex]

[tex]\omega^2+\omega+1=\dfrac{-1}{2}-i\dfrac{\sqrt{3}}{2}+\dfrac{-1}{2}+i\dfrac{\sqrt{3}}{2}+1=-1+1=0[/tex]

I hope to have helped! :)

Baltuilhe
 
Posts: 35
Joined: Fri Dec 14, 2018 3:55 pm
Reputation: 26

Re: Need help with few questions

Postby sunny » Sun Jul 28, 2019 2:13 am

I don't know what to say. I am speechless. I am not able to understand a word you wrote. I am new to this, is there an easy way to solve it? Or you think I must Understand what you wrote.

Good night, thanx for trying to help. Hoping someone can help me with a easier way.

I have not studied about cube root of unity, so I have no idea about it. I am learning determinants only at this point, can this be solved using determinants.

I have studied Order 2 and Just started reading creamer's rule.

sunny
 
Posts: 5
Joined: Sun Jul 28, 2019 12:42 am
Reputation: 0

Re: Need help with few questions

Postby sunny » Sun Jul 28, 2019 2:29 am

I might have found an answer, please let me know if I am right.

After searching google I have came across a few help pages, that says its already proven that

[tex]W^{2}[/tex] + W+1 =0

so since my answer is

[tex]W^{2}[/tex] + W

hence using above theorem we can say that my answer is -1


Am i thinking it correctly?


PS: I see that you have used symbol for unity, I can only see elected symbols in LaTeX Help, how can I find rest of the symbols?

sunny
 
Posts: 5
Joined: Sun Jul 28, 2019 12:42 am
Reputation: 0

Re: Need help with few questions

Postby sunny » Sun Jul 28, 2019 4:21 am


Solution to problem found so removing the post
sunny
 
Posts: 5
Joined: Sun Jul 28, 2019 12:42 am
Reputation: 0


Return to Algebra



Who is online

Users browsing this forum: Google Adsense [Bot] and 3 guests