Forgetting the middle term! Remember (a + bi)² = a² + 2abi + (bi)² = a² + 2abi - b². Dont just square the a and bi terms separately.
Always convert √(-a) to i√a first, where a is a positive real number. This prevents confusion and incorrect application of square root properties.
Convert to polar form (r, θ) and use the properties: z1 * z2 = (r1*r2, θ1 + θ2) and z1 / z2 = (r1/r2, θ1 - θ2). This simplifies calculations significantly.
Always consider the quadrant of the complex number in the Argand diagram. Use tan⁻¹(b/a) to find the reference angle, then adjust based on the quadrant to get the principal argument.
Multiply both the numerator and denominator by the conjugate of the denominator. This eliminates the imaginary part from the denominator, making simplification easier.
Remember that the conjugate of a + bi is a - bi. Only the sign of the imaginary part changes. This is crucial for division and finding moduli.
De Moivres Theorem states: (cos θ + i sin θ)^n = cos(nθ) + i sin(nθ). Remember to multiply the angle θ by the power n.
Equate the real and imaginary parts separately. If a + bi = c + di, then a = c and b = d. This allows you to form a system of equations.
Substitute your solution back into the original equation to verify that it satisfies the equation. For geometric interpretations, sketch the points on the Argand diagram.
Use the formula: z_k = r^(1/n) * [cos((θ + 2πk)/n) + i sin((θ + 2πk)/n)], where k = 0, 1, 2, ..., n-1. Remember to find all n distinct roots.