What is the quadratic formula used to solve ax^2 + bx + c = 0?

The quadratic formula is a powerful tool used to find the solutions for any quadratic equation in the standard form ( ax^2 + bx + c = 0 ), where ( a ), ( b ), and ( c ) are coefficients. The correct formula, ( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} ), provides the values of ( x ) for which the quadratic equation holds true.

This formula is derived from completing the square on the general quadratic equation. The ( \pm ) sign indicates that a quadratic equation can have two solutions: one that accounts for the positive square root and one for the negative square root. The term ( b^2 - 4ac ) is known as the discriminant and it determines the nature and number of the roots. If the discriminant is positive, there are two real and distinct solutions. If it is zero, there is exactly one real solution (a repeated root). If the discriminant is negative, there are two complex solutions.

The clarity of the formula comes from its structure: the numerator consists of the negative of the linear coefficient ( b ) and the square root of the discriminant,