Find all complex numbers z such that z2 1 2 sqrt6 i Soluti

Find all complex numbers z such that z² = -1 + 2 sqrt(6) i.

:(

Solution

Let z = a + bi

Substitute into given equation: (a + bi)² = -1 + 2 sqrt(6) i

Expand: a² - b² + 2 ab i = - 1 + 2 sqrt(6) i

Real part and imaginary parts must be equal.

a² - b² = - 1 and 2 ab = 2 sqrt(6)

Equation 2 ab = 2 sqrt(6) gives: b = sqrt(6) / a

Substitute: a² - ( sqrt(6) / a )²) = - 1

a⁴ - 6 = - a²

Solve above equation and select only real roots: a = sqrt(2) and a = - sqrt(2)

Substitute to find b and write the two complex numbers that satisfies the given equation.

z1 = sqrt(2) + sqrt(3) i ,

z2 = - sqrt(2) - sqrt(3) i