If z be a complex number satisfying z2 + z + 1 = 0. Find the value of |z|.
Let z = x + iy
Substitute z in given equation
x2 - y2 + 2xyi + x + iy +1 = 0
(x2 - y2 + x +1) + y(2x + 1)1 = 0
Comparing real and imaginary part of the above given equation
x2 - y2 + x +1 = 0----------------(1) or
y(2x + 1) = 0 ---------(2)
y = 0,2x + 1=0
Substitute value of y in equation 1When 2x+1=0,x=−12when y=014−y2+12=0x2+x+1=0y2=34x=−1+––√−32y=+––√32,x=−12x=−1+––√3i2
Substitute the value of x and y to get the complex number z.
Case 1: when y = 0
Complex number
z = x + iy
= −1+––√3i2
Case 2: when x = −12, y = +––√32
z = x + iy
= −1+––√3i2
|z| = √14+34 = √1 = 1
So, modulus of z = 1.