Quadratic equations can be solved by graphing, using the quadratic formula, completing the square, and factoring. What are the pros and cons of each of these methods? When might each method be most appropriate? Which method do you prefer?

By the way, as far as I am concerned, using the quadratic equation is just another way of saying "complete the square".

Quadratic formula: In my opinion, this is the most general method, and likely the best overall. It will always work, and if you memorize the formula, there is no guessing about how to apply it. The formula allows you to find real and complex solutions.

Graphing: graphing the equation will only give valid results if the equation has real solutions. The solutions are located where the graph crosses the x axis. If the solutions are irrational or fractions with large denominators, this method will only be able to approximate the solutions. If you have a graphing calculator, this method is the quickest. If you don't have a calculator, it can be tedious and difficult to graph the equation, and you have to be very precise when you do it.
Completing the square: This is probably the most difficult method. I find it hardest to remember how to apply this method. Since the quadratic formula was derived from this method, I don't think there is a good reason to use completing the square when you have the formula (and are allowed to use it).
Factoring: this is probably the easiest method for solving an equation with integer solutions. If you can see how to split up the original equation into its factor pair, this is the quickest and allows you to solve the problem in one step.

The different methods for solving quadratic equations have their own distinct pros and cons. Let's explore each method and discuss their advantages and disadvantages, as well as the situations in which they might be most appropriate.

1. Graphing:
Pro: Graphing visually represents the equation, making it easier to understand the behavior of the quadratic function.
Con: Graphing can be imprecise, particularly when trying to find exact solutions.
Appropriate: Graphing is useful when you want to gain a general understanding of the roots and shape of the quadratic equation, but it might not be the most efficient method for finding precise solutions.

2. Quadratic Formula:
Pro: The quadratic formula always provides accurate solutions for any quadratic equation.
Con: It may involve complex calculations, particularly when dealing with equations that have large coefficients.
Appropriate: Using the quadratic formula is most appropriate when you need precise solutions for any quadratic equation, regardless of the coefficient values.

3. Completing the Square:
Pro: Completing the square helps in converting a quadratic equation into its standard form and provides a geometrical interpretation.
Con: The process of completing the square can involve tedious calculations.
Appropriate: Completing the square is useful when you need the equation in vertex form or when solving quadratic equations with integers, as it often yields whole number solutions.

4. Factoring:
Pro: Factoring allows you to find the roots of a quadratic equation quickly when it is easily factorable.
Con: Not all quadratic equations can be factored easily, especially those with irrational or complex roots.
Appropriate: Factoring is most appropriate when the quadratic equation is factorable, as it provides an efficient method for finding the roots.

The preference for a particular method depends on the specific scenario. If you prioritize accuracy and are comfortable with complex calculations, the quadratic formula is a reliable choice. If you prefer a visual representation to understand the equation, graphing may suit you. If simplicity is important, factoring may be preferred when it is possible. Completing the square is useful for obtaining vertex form or solving equations with whole number solutions.

Ultimately, the method you choose depends on your personal preferences, the complexity of the equation, and the level of accuracy you require.