Quadratic Formula
The quadratic formula gives the solutions to any quadratic equation $ax^2 + bx + c = 0$ as $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
Formula
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
Definition
The quadratic formula solves any quadratic equation $ax^2 + bx + c = 0$ directly: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where the "$\pm$" produces both solutions at once (or one, if the discriminant is zero). It is derived by completing the square on the general form, so understanding that process reveals where the formula comes from rather than requiring rote memorization. Over $\mathbb{C}$, it always yields two values (counting multiplicity), and it generalizes to closed-form solutions for the cubic (Cardano) and quartic (Ferrari), though no analogous formula exists for degree $5$ and above (the Abel-Ruffini theorem).
Example
$x^2 - 5x + 6 = 0$: $a=1$, $b=-5$, $c=6$, so $x = (5 \pm \sqrt{25-24})/2 = (5 \pm 1)/2$, giving $x = 3$ or $x = 2$. $3x^2 + 2x - 1 = 0$: $x = (-2 \pm \sqrt{4+12})/6 = (-2 \pm 4)/6$, giving $x = 1/3$ or $x = -1$. For $x^2 + x + 1 = 0$, $\Delta = 1 - 4 = -3 < 0$, giving $x = (-1 \pm i\sqrt{3})/2$, the primitive $6$th roots of unity satisfying $x^6 = 1$.
Key Insight
The quadratic formula always works, even when factoring is impossible, which is why it is worth memorizing; but it is really just completing the square carried out once, in general. As a nested square-root expression, its existence is exactly what makes degree $2$ solvable by radicals, and the impossibility of an analogous formula for degree $5$ and above is one of the deepest results in algebra, proved using group theory.