How do you find all the real and complex roots of # x^4=1#?

Question

25249 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-05-05T16:27:05

Real roots of the equation are #1# and #-1#

and complex roots are #i# and #-i#.

As #x^4=1#, #(x^4-1)=0#

Using #a^2-b^2=(a+b)(a-b)# above is equal to

#(x^2+1)(x^2-1)=0#

or #(x^2-i^2)(x^2-1)#, (as #i^2=-1#)

or #(x+i)(x-i)(x+1)(x-1)=0#

Hence real roots of the equation are #1# and #-1#

and complex roots are #i# and #-i#.