Discriminant
AlgebraThe discriminant (b^2 - 4ac) of a quadratic equation determines whether it has two real solutions, one real solution, or no real solutions.
Formula
D = b^2 - 4ac
Definition
The discriminant is the part of the quadratic formula under the square root: $b^2 - 4ac$. It tells you how many real solutions a quadratic equation has without fully solving it.
Example
For $x^2 - 4x + 4 = 0$: $D = 16 - 16 = 0$. One solution. For $x^2 - 5x + 6 = 0$: $D = 25 - 24 = 1 > 0$. Two solutions.
Key Insight
Positive discriminant = $2$ real solutions. Zero discriminant = $1$ real solution. Negative discriminant = $0$ real solutions. You can predict the answer count before solving.
Definition
The discriminant $D = b^2 - 4ac$ of $ax^2 + bx + c = 0$ determines the nature of the roots: $D > 0$ means two distinct real roots, $D = 0$ means one repeated real root (double root), $D < 0$ means two complex conjugate roots (no real solutions). When $D$ is a perfect square and $a$, $b$, $c$ are integers, the roots are rational.
Example
$2x^2 + 3x + 5 = 0$: $D = 9 - 40 = -31 < 0$. No real solutions. The parabola $y = 2x^2 + 3x + 5$ lies entirely above the x-axis.
Key Insight
The discriminant is the key diagnostic tool for quadratics. In real-world models, $D < 0$ means the modeled situation never reaches zero (e.g., a ball that does not fall to the ground in the modeled region).
Definition
The discriminant of a general polynomial $f$ of degree $n$ is $\Delta(f) = a_n^{2n-2} \prod_{i < j} (r_i - r_j)^2$, where $r_i$ are the roots of $f$ in $\mathbb{C}$. For the quadratic $ax^2 + bx + c$, this reduces to $b^2 - 4ac$. The discriminant is zero iff $f$ has a repeated root (equivalently, $\gcd(f, f')$ is not constant). For a cubic $x^3 + px + q$, the discriminant is $-4p^3 - 27q^2$.
Example
For $x^3 - 3x + 2 = 0$: discriminant $= -4(-3)^3 - 27(2)^2 = 108 - 108 = 0$. The cubic has a repeated root; indeed $x^3 - 3x + 2 = (x-1)^2(x+2)$.
Key Insight
The discriminant is a polynomial in the coefficients of $f$. It is invariant under field extensions: $\Delta(f) = 0$ in any field containing the coefficients iff $f$ has a repeated root over the algebraic closure. This makes it a fundamental tool in Galois theory and algebraic number theory.