How can the roots of a quadratic equation be found?

Finding the roots of a quadratic equation can be accomplished through several methods, which is precisely why the answer that includes factoring, completing the square, or using the quadratic formula is accurate.

Factoring involves expressing the quadratic in a form where it can be set to zero, making it easy to solve for the variable. Completing the square is a technique that transforms the quadratic equation into a perfect square trinomial, allowing for straightforward resolution of the equation. The quadratic formula, derived from the standard form of a quadratic equation, provides a direct way to calculate the roots regardless of whether the equation can be factored easily.

While graphing can visually demonstrate the roots by pinpointing where the graph intersects the x-axis, it does not provide the roots in a numerical form or is considered an exact method to find values. Finding the derivative of the equation relates to understanding the rate of change and optimization rather than solving for the roots of the quadratic. Multiplying the equation by zero does not yield usable information about the roots, as it simply leads to a trivial equation of 0 = 0. Each of these alternative methods serves different purposes and does not directly lead to determining the roots of a quadratic in the same effective manner as the first option.