Solution of Equations

Quadratic Equations with Complex Coefficients

Understand that quadratic equations with complex coefficients can have solutions in the form a ± bi, where a and b are real numbers.
Apply the quadratic formula, x = [-b ± sqrt(b² - 4ac)] / 2a, where a, b and c are coefficients of the quadratic equation ax² + bx + c = 0. If b² - 4ac is negative, the solutions will be complex numbers.
Realise that the discriminant (b² - 4ac) of the quadratic equation determines the nature of its roots. If it’s negative, the roots are complex conjugates of each other.

Cubic and Quartic Equations

Acknowledge that cubic and quartic equations can also have complex roots, even if all coefficients are real.
Note that the solutions to quartic equations can be found by solving a resolvent cubic. Be sure to check all possible solutions.
Realise that if a polynomial has real coefficients, any complex roots must appear in conjugate pairs.

Solution by Iteration (Solving Equations using Complex Analysis)

Understand the method of Newton-Raphson iteration for finding approximate roots of complex equations. It uses the derivative of the function to estimate the change from the initial guess to the solution.
Note that when applying iteration to complex tasks, the iteration may not converge if the initial guess is far from the actual solution or if the function is not well-behaved.
Remember to always give your answer in the a + bi form for complex solutions.

Fundamental Theorem of Algebra

Know the Fundamental Theorem of Algebra, which states that a polynomial of degree n will have exactly n roots (including multiple roots and complex roots).
Grasp that the Fundamental Theorem of Algebra does not provide a method for finding the roots of a polynomial, but rather ensures us that solutions do exist.
Realise that the process of finding complex roots often involves algebraic manipulation, use of relationships between roots, and knowledge of techniques for solution.

Remember to check your answers and ensure you are confident in your method before moving on. Understanding complex numbers and the solution of equations within the context of the topic is critical for success. It’s not just about getting the right answer, but truly understanding the processes that led you there.