Question

x^3 - i + sqrt3 = 0. Solve for x.

Answer 1

the equation is
x^3 -i + sqrt(3) = 0

does it say how many solutions it must have?

Answer 2

x^3 = i - sqrt(3)

x = cubrt ( i - sqrt(3))

Answer 3

@perl No it doesn't specify how many.

Answer 4

the cube root of a complex number has three solutions, thats why i was wondering

Answer 5

@perl Is that the final answer?

Answer 6

so we want
x = (-sqrt(3) + i ) ^(1/3)

lets use demoivre's theorem to get all three solutions

Answer 7

first change -sqrt(3) + i to polar trig form

Answer 8

so
r = sqrt( (-sqrt(3))^2 + 1^2 ) = sqrt( 3 + 1 ) = 2

theta = atan ( -sqrt(3), 1 )

Answer 9

people also use arctan, i prefer the atan function since it tells you the exact angle

Answer 10

the angle is 5pi/6
http://www.wolframalpha.com/input/?i=atan%28-sqrt%283%29%2C1%29

Answer 11

so we have so far
x = [2 ( cos (5pi/6) + i sin(5pi/6) ] ^(1/3)

Answer 12

x = [2 ( cos (5pi/6 + 2pi*n) + i sin(5pi/6 + 2pi*n) ) ] ^(1/3)

now use demoivres theorem

x = [ 2^(1/3) ( cos ( 5pi/6* 1/3 + 2pi/3 * n ) + i ( sin5pi/6 * 1/3 + 2pi/3*n ) ]

Answer 13

let n = 0,1,2

Answer 14

this gives us 3 solutions

x = [2^(1/3)(cos( 5pi/18 + 2pi/3* 0) + i( sin5pi/18 + 2pi/3*0 ))]

x= [2^(1/3)(cos( 5pi/18 + 2pi/3*1) + i( sin5pi/18 + 2pi/3*1 ))]

x = [2^(1/3)(cos( 5pi/18 + 2pi/3* 2) + i( sin5pi/18 + 2pi/3*2 ))]

Answer 15

thats a long expression, which we can estimate

Answer 16

x = .80986 + .9651555 i
x = -1.24078 + .21878 i
x = .430918 - 1.1839

Answer 17

which agrees with wolfram
http://www.wolframalpha.com/input/?i=solve+x^3+-+i+%2B+sqrt3+%3D+0