Question

= 27. then find the value of x3
Fo+b=12 and ab = 27, then find the value of
- B3
.
1
3
X
F 3x - 2y = 11 and xy = 12 then find the value
of 27x2-8y?
² o²b + a62​

Answer 1

Answer:

In order to find the 3 roots of the equation, let z be the complex variable such that z = x + 2 and z = r(cos@ + i sin@)

Therefore, z^3 = r^3 (cos@ + i sin@)^3 = r^3 (cos3@ + i sin3@) = 27, which implies:

r^3 = 27 and r = 3, and

cos3@ + i sin3@ = 1, which implies cos3@ = 1 and sin3@ = 0 and therefore 3@ = 2npi and @ = 2npi/3, where n = 0, 1,2

For n = 0, @ = 0 and z = 3(cos0 + i sin0) = 3 and therefore, x = 3 - 2 = 1

For n = 1, @ = 2pi/3 and z = 3[cos(2pi/3) + i sin(2pi/3)] = 3(-1/2 + i sqrt3/2)

=-3/2 + i 3sqrt.3/2 and x = -3/2 + i 3sqrt.3/2 - 2 = -7/2 + i 3sqrt.3/2

For n = 2, @ = 4pi/3 and z = 3[cos(4pi/3) + i sin(4pi/3)] = 3(-1/2 - sqrt.3/2)

= -3/2 - 3sqrt.3/2 and x = -3/2 - 3sqrt.3/2 - 2 = -7/2 - i 3sqrt.3/2

The solution set ={1, (-7/2 + i 3sqrt.3/2), (-7/2 - i 3sqrt.3/2)}