Understand the notation and concept of a complex number i, where i2 equals -1, and every complex number can be written as a + bi, with a and b being real numbers. Use i2 equals -1 and mathematical properties to perform operations with complex numbers. Find the conjugate of a complex number and use it to find moduli and quotients. Represent complex numbers on the complex plane in rectangular and polar form, acknowledging both represent the same number.

Perform addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane, using properties for computation. Calculate the distance between numbers in the complex plane as the modulus of the difference. Determine the midpoint of a segment as the average of the numbers at its endpoints. Solve quadratic equations with real coefficients that yield complex solutions. Expand polynomial identities to complex numbers. Understand the Fundamental Theorem of Algebra and prove its validity for quadratic polynomials.