quadratic formula

The quadratic formula is a guaranteed, plug-and-play recipe that solves any quadratic equation, even ones that refuse to factor nicely. You feed in the three coefficients a, b, c and it hands back the roots. No cleverness required — it always works.

For ax^2 + bx + c = 0 with a ≠ 0, the roots are x = (-b ± sqrt(b^2 - 4ac)) / (2a). The ± means you take two answers, one with the plus and one with the minus, which is why a quadratic can have two roots. The expression under the root, b^2 - 4ac, is the discriminant, and it tells you in advance how many real roots there will be.

The formula is not magic from nowhere — it is exactly what completing the square produces when carried out on the general equation. A practical caution: be careful with signs, divide the whole numerator by 2a (not just part of it), and remember that if b^2 - 4ac is negative the roots are complex, not real.