Quadratic Formula
AlgebraThe quadratic formula gives the solutions to any quadratic equation ax^2 + bx + c = 0 as x = (-b +/- sqrt(b^2 - 4ac)) / (2a).
Formula
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
Definition
The quadratic formula solves any quadratic equation. If you have $ax^2 + bx + c = 0$, the solutions are $x = (-b \pm \sqrt{b^2 - 4ac})/(2a)$. The "$\pm$" means you get two answers.
Example
$x^2 - 5x + 6 = 0$: $a = 1$, $b = -5$, $c = 6$. $x = (5 \pm \sqrt{25 - 24})/2 = (5 \pm 1)/2$. So $x = 3$ or $x = 2$.
Key Insight
The quadratic formula always works for any quadratic equation, even when factoring is impossible. It is worth memorizing.
Definition
The quadratic formula $x = (-b \pm \sqrt{b^2 - 4ac})/(2a)$ gives both solutions of $ax^2 + bx + c = 0$ simultaneously. The $\pm$ produces two solutions (or one if the discriminant is zero). The formula is derived by completing the square on the general form $ax^2 + bx + c = 0$.
Example
$3x^2 + 2x - 1 = 0$: $a=3$, $b=2$, $c=-1$. $x = (-2 \pm \sqrt{4+12})/6 = (-2 \pm 4)/6$. So $x = 2/6 = 1/3$ or $x = -6/6 = -1$.
Key Insight
The quadratic formula is derived by completing the square, so understanding completing the square gives you insight into where the formula comes from rather than just memorizing it.
Definition
The quadratic formula is the explicit closed-form solution of the degree-$2$ polynomial equation. It is derived by completing the square: $ax^2 + bx + c = 0$ implies $(x + b/(2a))^2 = (b^2 - 4ac)/(4a^2)$, giving $x = (-b \pm \sqrt{b^2 - 4ac})/(2a)$. Over $\mathbb{C}$, the formula always yields two values (counting multiplicity). The formula generalizes to the cubic (Cardano) and quartic (Ferrari) but no analogous formula exists for degree $\ge 5$ (Abel-Ruffini theorem).
Example
For $x^2 + x + 1 = 0$: $\Delta = 1 - 4 = -3 < 0$. Solutions: $x = (-1 \pm i\sqrt{3})/2$, the primitive $6$th roots of unity, satisfying $x^6 = 1$.
Key Insight
The quadratic formula is a depth-$5$ arithmetic expression (nested square root). Its existence is what makes degree $2$ solvable by radicals. The impossibility for degree $\ge 5$ is one of the deepest theorems in algebra, proved using group theory.