The Fundamental Theorem Of Algebra
Any polynomial of degree with complex coefficient has (complex) zeros, provided we count the zeros with their multiplicities.
In particular, from The Fundamental Theorem f Algebra follows that any quadratic polynomial with complex coefficients has two complex zeros. This particular case of the FTA is not difficult to prove by employing the Quadratic Formula, provided we can prove that there are two values of the square root of any (nonzero) complex number. Your Mini-Project is particular case of this statement which is true generally.
Problem Find the square roots of -5+12i, i.e. solve the equation z^2= -5+12i by setting the system of equations for the real part x and the imaginary part Y of the solution z = x+yi. Remember that X and Yare real numbers.