Which formula is used to find the roots of a quadratic equation?

The formula used to find the roots of a quadratic equation, which is generally expressed in the form ( ax^2 + bx + c = 0 ), is derived from a method known as completing the square. The correct formula is ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ).

In this formula, ( -b ) represents the negation of the coefficient of the ( x ) term, while ( \sqrt{b^2 - 4ac} ) calculates the discriminant. The discriminant is crucial because it determines the nature of the roots. If it is positive, there are two distinct real roots; if it is zero, there is exactly one real root (a repeated root); and if it is negative, it indicates two complex roots.

Furthermore, the denominator ( 2a ) normalizes the roots with respect to the leading coefficient of the quadratic equation, ensuring that we account for the coefficient of ( x^2 ) when finding the solutions.

The other options provided do not yield the correct roots for a quadratic equation. For instance, alternative formulations either miscalculate the discriminant or incorrectly represent the coefficients, which would lead