Which method can be used to solve quadratic equations?

Quadratic equations can be solved using various methods, making the choice of methodologies both practical and versatile. The correct answer includes three distinct methods: factoring, completing the square, and using the quadratic formula.

Factoring involves expressing the quadratic equation in the form of a product of two binomials. This can often lead to quick solutions, especially when the roots are rational numbers.

Completing the square is another powerful technique that transforms the quadratic equation into a perfect square trinomial. This method is particularly useful because it not only allows for solving the equation but also gives insight into the vertex form of the quadratic function, helping to identify the vertex of the parabola.

The quadratic formula, which is derived from the standard form of a quadratic equation, can solve any quadratic equation, regardless of whether the roots are rational, irrational, or complex. The formula is ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ) and provides a reliable method to find roots when factoring is not practical.

Each of these methods has its advantages depending on the specific equation and context, which is why the combination of all three is the most comprehensive approach to solving quadratic equations.