Quadratic Equation
A quadratic equation is a polynomial equation of degree $2$, generally written as $ax^2 + bx + c = 0$, solvable by factoring, completing the square, or the quadratic formula.
Formula
ax^2 + bx + c = 0
Definition
A quadratic equation is an equation where the highest power of the variable is $2$, written in standard form $ax^2 + bx + c = 0$ with $a$ not zero; it can be solved by factoring, completing the square, or the quadratic formula, and the number of real solutions is determined by the discriminant $b^2 - 4ac$. By the Fundamental Theorem of Algebra, every quadratic equation has exactly $2$ roots in $\mathbb{C}$ (counted with multiplicity); by Vieta's formulas the sum of the roots is $-b/a$ and their product is $c/a$, and the discriminant $\Delta = b^2 - 4ac$ tells which kind of roots occur: $\Delta > 0$ gives two distinct real roots, $\Delta = 0$ gives one repeated real root, and $\Delta < 0$ gives two complex conjugate roots.
Example
$x^2 - 5x + 6 = 0$ factors as $(x-2)(x-3) = 0$, giving $x = 2$ or $x = 3$. $x^2 + 2x - 8 = 0$ factors as $(x+4)(x-2) = 0$, giving $x = -4$ or $x = 2$. For $x^2 + 1 = 0$, $\Delta = -4 < 0$, so the roots $x = \pm i$ are complex conjugates: no real solutions, but two solutions in $\mathbb{C}$.
Key Insight
The "quad" in quadratic comes from the Latin for square, since the variable is squared, and a quadratic equation can have up to two answers. Every quadratic can be solved by the quadratic formula even when factoring is difficult, so knowing all three methods, factoring, completing the square, and the formula, gives real flexibility. Quadratics are the simplest case where the Fundamental Theorem of Algebra becomes visible and where complex numbers first become necessary; the quadratic formula itself was known to Babylonian mathematicians around 2000 BCE.