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Discriminant Calculator

Calculate the discriminant (b²-4ac) and instantly determine whether a quadratic equation has two real roots, one repeated root, or complex roots.

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Standard Form

Classic quadratic x² − 5x + 6 = 0 with two integer roots (x = 2 and x = 3)

Key values: a = 1 · b = −5 · c = 6

Physics Problem

Projectile height equation −4.9t² + 20t + 1.5 = 0 (when does the ball hit the ground?)

Key values: a = −4.9 · b = 20 · c = 1.5

Complex Roots

Equation x² + 2x + 5 = 0 with no real solutions (discriminant < 0)

Key values: a = 1 · b = 2 · c = 5

Documentation

The Discriminant

The discriminant of a quadratic $ax^2 + bx + c$ is:

\Delta = b^2 - 4ac

It tells you everything about the nature of the roots without computing them. It's the quantity under the square root in the quadratic formula.

Three Cases

Condition	Root nature	Parabola behavior	Example
$\Delta > 0$	Two distinct real roots	Crosses x-axis at two points	$x^2 - 5x + 6$ , Δ = 1
$\Delta = 0$	One repeated root (double root)	Touches x-axis at vertex	$x^2 - 6x + 9$ , Δ = 0
$\Delta < 0$	Two complex conjugate roots	Never touches x-axis	$x^2 + 1$ , Δ = −4

Perfect square test: When $\Delta > 0$ and is a perfect square, the roots are rational (factorable over integers). When $\Delta > 0$ but not a perfect square, the roots are irrational (involve square roots).

Applications of the Discriminant

Line-circle intersection: Substituting a line equation into a circle equation gives a quadratic. $\Delta > 0$ : two intersection points (secant). $\Delta = 0$ : tangent line. $\Delta < 0$ : no intersection.
Optimization constraints: Ensuring a quadratic expression is always positive requires $a > 0$ and $\Delta < 0$ .
Parameter ranges: “For which values of $k$ does $x^2 + kx + 4 = 0$ have real roots?” Solve $k^2 - 16 \geq 0$ to get $|k| \geq 4$ .

Beyond Quadratics

The discriminant generalizes to higher-degree polynomials. For a cubic $ax^3 + bx^2 + cx + d$ :

\Delta = 18abcd - 4b^3d + b^2c^2 - 4ac^3 - 27a^2d^2

$\Delta > 0$ : three distinct real roots. $\Delta = 0$ : repeated root. $\Delta < 0$ : one real root and two complex conjugate roots.

Frequently Asked Questions

What is the discriminant of a quadratic equation?

The discriminant is the expression $b^2 - 4ac$ from the quadratic formula. It appears under the square root and determines the nature of the roots without solving the equation. For $ax^2 + bx + c = 0$ , calculate $b^2 - 4ac$ to classify the solutions.

What does a positive discriminant mean?

A positive discriminant ( $b^2 - 4ac > 0$ ) means the equation has two distinct real roots. If the discriminant is also a perfect square, the roots are rational numbers and the quadratic can be factored over the integers. If not a perfect square, the roots are irrational.

What does a zero discriminant mean?

A zero discriminant ( $b^2 - 4ac = 0$ ) means the equation has exactly one repeated (double) root. Graphically, the parabola touches the x-axis at exactly one point, which is its vertex. The root is $x = -b/(2a)$ .

What does a negative discriminant mean?

A negative discriminant ( $b^2 - 4ac < 0$ ) means there are no real roots. The equation has two complex conjugate roots of the form $a + bi$ and $a - bi$ . Graphically, the parabola never crosses the x-axis.

How is the discriminant used outside of solving equations?

The discriminant determines line-circle intersections (positive = secant, zero = tangent, negative = no intersection), checks if a quadratic expression is always positive or negative, and finds parameter ranges that guarantee real solutions. It generalizes to cubic and higher-degree polynomials.

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