Quadratic Equation
AlgebraA 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 $2$. It can have two solutions, one solution, or no real solutions.
Example
$x^2 - 5x + 6 = 0$. Factor: $(x-2)(x-3) = 0$. Solutions: $x = 2$ or $x = 3$. Check: $4 - 10 + 6 = 0$ and $9 - 15 + 6 = 0$. Both work.
Key Insight
The "quad" in quadratic comes from the Latin for square, because the variable is squared. A quadratic can have up to two answers.
Definition
A quadratic equation has the standard form $ax^2 + bx + c = 0$ where $a$ is not zero. It can be solved by factoring (if factorable), completing the square, or using the quadratic formula. The number of real solutions is determined by the discriminant $b^2 - 4ac$.
Example
$x^2 + 2x - 8 = 0$: factor as $(x+4)(x-2) = 0$. Solutions: $x = -4$ or $x = 2$. Both satisfy the original equation.
Key Insight
Every quadratic equation can be solved by the quadratic formula, even when factoring is difficult or impossible. Knowing all three methods (factor, complete the square, formula) gives you flexibility.
Definition
A quadratic equation $ax^2 + bx + c = 0$ ($a \neq 0$) has its solution set characterized by the Fundamental Theorem of Algebra: exactly $2$ roots in $\mathbb{C}$ (counted with multiplicity). By Vieta's formulas, the sum of roots is $-b/a$ and the product is $c/a$. The discriminant $\Delta = b^2 - 4ac$ determines the nature of roots: $\Delta > 0$ gives two distinct real roots, $\Delta = 0$ gives one repeated real root, $\Delta < 0$ gives two complex conjugate roots.
Example
$x^2 + 1 = 0$: $\Delta = -4 < 0$. Roots: $x = \pm i$ (complex conjugates). In $\mathbb{R}$, no solutions; in $\mathbb{C}$, two solutions.
Key Insight
Quadratic equations are the simplest non-trivial case where the Fundamental Theorem of Algebra is visible and where complex numbers first become necessary. The quadratic formula was known to Babylonian mathematicians around 2000 BCE.