Discriminant Calculator

Calculate the discriminant of a quadratic equation ax2 + bx + c = 0 to determine the number and type of roots.

Enter Coefficients (ax2 + bx + c = 0)

Result

Discriminant (Delta)
1
Two distinct real roots
Discriminant1
Nature of RootsTwo distinct real roots
Root 1 (x1)3
Root 2 (x2)2
Sum of Roots (-b/a)5
Product of Roots (c/a)6

Step-by-Step Solution

Delta = b^2 - 4ac = (-5)^2 - 4(1)(6) = 25 - 24 = 1

What Is the Discriminant?

The discriminant is a value calculated from the coefficients of a quadratic equation ax2 + bx + c = 0. It is denoted by the Greek letter Delta and defined as b2 - 4ac. The discriminant reveals crucial information about the nature of the roots of the quadratic equation without actually solving it.

The discriminant appears under the square root sign in the quadratic formula: x = (-b +/- sqrt(b2 - 4ac)) / (2a). Since you cannot take the square root of a negative number within the real numbers, the sign of the discriminant determines whether the roots are real or complex.

Interpreting the Discriminant

Delta > 0: Two Distinct Real Roots

The parabola crosses the x-axis at two different points. The roots are real and unequal.

x = (-b +/- sqrt(Delta)) / 2a

Delta = 0: One Repeated Real Root

The parabola touches the x-axis at exactly one point (the vertex). The root is a double root.

x = -b / 2a (repeated)

Delta < 0: Two Complex Conjugate Roots

The parabola does not intersect the x-axis. The roots are complex conjugates of each other.

x = (-b +/- i*sqrt(|Delta|)) / 2a

Perfect Square Discriminants

When the discriminant is a perfect square (and the coefficients are integers), the quadratic can be factored over the integers. For example, x2 - 5x + 6 = 0 has discriminant 1, which is a perfect square, and it factors as (x - 2)(x - 3) = 0. This is an important concept in algebra for determining whether a quadratic is factorable by simple methods.

Discriminant and the Graph

The discriminant also provides information about the graph of y = ax2 + bx + c:

  • Delta > 0: The parabola has two x-intercepts.
  • Delta = 0: The parabola has one x-intercept (vertex on x-axis).
  • Delta < 0: The parabola has no x-intercepts (entirely above or below x-axis).

Vieta's Formulas

For a quadratic ax2 + bx + c = 0 with roots r1 and r2:

  • Sum of roots: r1 + r2 = -b/a
  • Product of roots: r1 * r2 = c/a
  • Difference of roots: |r1 - r2| = sqrt(Delta) / |a|

These relationships are independent of the quadratic formula and provide a way to verify solutions or construct equations from known roots.

Examples

  • 2x2 + 4x + 2 = 0: Delta = 16 - 16 = 0. One repeated root: x = -1.
  • x2 + x + 1 = 0: Delta = 1 - 4 = -3. Two complex roots: x = (-1 +/- i*sqrt(3)) / 2.
  • 3x2 - 12x + 9 = 0: Delta = 144 - 108 = 36. Two real roots: x = 3, x = 1.
  • x2 - 2x + 1 = 0: Delta = 4 - 4 = 0. Repeated root: x = 1 (perfect square trinomial).

Applications of the Discriminant

The discriminant is used in many areas of mathematics and science:

  • Optimization: Determining whether constraints have feasible solutions.
  • Physics: Analyzing projectile motion, determining when objects meet.
  • Engineering: Stability analysis of systems described by quadratic equations.
  • Computer graphics: Ray-sphere intersection tests use the discriminant.