Question

Name three ways to find the solution(s) of a quadratic equation. Which way do you prefer? Explain.

Short answer

The three ways to solve a quadratic equation include Factoring, Using the Quadratic Formula, and Completing the Square. The preference in this discussion is for the Quadratic Formula, as it's always applicable and directly solves for the roots, whether real or complex.

Step-by-step solution

Identifying Methods of Solving Quadratic Equations

There are primarily three ways to solve a quadratic equation: 1. Factoring, 2. Using the Quadratic Formula, and 3. Completing the Square.

Elaborate on Factoring Method

Factoring is the method of converting a quadratic equation into a product of factors, letting each factor equal zero and then solving for the variable. It is simple and quick, but it may not always be possible if the equation cannot be easily factored.

Elaborate on Quadratic Formula Method

The Quadratic Formula, \( x= \frac{-b ± \sqrt{b^2-4ac}}{2a} \), is used when a quadratic equation cannot be easily factored. It will always provide a solution, whether the roots are real or complex.

Elaborate on Completing the Square Method

Completing the square involves re-arranging the equation to create a perfect square trinomial on one side. After achieving this, the square root of both sides is taken to solve for the variable. It has the benefit of always providing a valid solution, but it can be a bit complex.

Personal Preference

Though it is subjective, let's give preference to the Quadratic Formula. This method is generally the best because it is always applicable and solves for the roots directly, whether they are real or complex. It can help bypass the challenge of factoring or completing the square, particularly when the equation can't be easily factored.