Discriminant Calculator
Calculate the discriminant of a quadratic, cubic or quartic polynomial. Choose your degree, enter the coefficients, and instantly see the discriminant value, the number and nature of the roots, the actual root values (quadratic), the vertex and axis of symmetry, and a plain-English explanation of what it all means.
Formula
Worked example
For x^2 - 5x + 6 = 0 (a=1, b=-5, c=6): Delta = (-5)^2 - 4(1)(6) = 25 - 24 = 1. Delta = 1 > 0 and is a perfect square, so two rational roots. Quadratic formula: x = (5 +/- 1) / 2, giving x1 = 3 and x2 = 2. Axis of symmetry: x = 5/2 = 2.5. Vertex: (2.5, -0.25). Parabola opens upward.
What the discriminant is and why it matters
The discriminant is the expression that sits under the square-root sign in the quadratic formula. For a quadratic ax^2 + bx + c = 0 it is Delta = b^2 - 4ac. Its sign tells you everything about the roots before you do any further algebra. A positive discriminant means two real roots exist and the parabola crosses the x-axis twice. A zero discriminant means the parabola is tangent to the x-axis, touching at exactly one point called the repeated root. A negative discriminant means the parabola never touches the x-axis at all and the two roots are a conjugate pair of complex numbers. For cubics the same principle holds but the formula is more involved: Delta = 18abcd - 4b^3d + b^2c^2 - 4ac^3 - 27a^2d^2. Positive means three distinct real roots; zero means a repeated root; negative means one real root and two complex conjugates.
Reading the sign of Delta
When Delta > 0 the square root in the quadratic formula is a real positive number, so the plus and minus branches give two distinct real answers. If Delta happens to be a perfect square (such as 1, 4, 9, 25) the square root is a whole number and both roots are rational, which means they can be written as exact fractions. If Delta is positive but not a perfect square the roots are irrational. When Delta = 0 the square root vanishes and both branches collapse to x = -b / 2a, a single rational number equal to the axis of symmetry. When Delta < 0 the square root of a negative number is imaginary, giving two complex conjugates: x = -b/(2a) +/- i*sqrt(-Delta)/(2a).
Vertex, axis of symmetry and parabola orientation
Every quadratic parabola has a unique line of symmetry at x = -b / 2a, known as the axis of symmetry. The vertex is the turning point of the parabola, located at (h, k) where h = -b / 2a and k = a*h^2 + b*h + c. When the leading coefficient a is positive the parabola opens upward and the vertex is a minimum; when a is negative it opens downward and the vertex is a maximum. Knowing the vertex and axis of symmetry lets you sketch the curve immediately, even before you find the roots. The discriminant tells you how many times that curve crosses the horizontal axis, and the vertex tells you how high or low the turning point is.
Cubic discriminant and its interpretation
For a cubic ax^3 + bx^2 + cx + d = 0 the discriminant is Delta = 18abcd - 4b^3d + b^2c^2 - 4ac^3 - 27a^2d^2. A positive value guarantees three distinct real roots; the cubic S-curve crosses the x-axis three times. A zero discriminant means a repeated root: either all three roots are equal (a triple root, the curve is tangent to the x-axis at its inflection point) or one root is simple and another is a double root (the curve touches but does not cross at the double root). A negative discriminant means only one real root and two complex conjugates. Unlike quadratics, there is no simple formula that gives you the cubic roots directly, but the discriminant alone classifies them unambiguously.
Using the discriminant in practice
Before solving a quadratic, compute the discriminant in one quick step to find out whether factoring will even produce rational answers. If Delta is a perfect square, rational factoring works and you can avoid the quadratic formula entirely. If Delta is positive but irrational, the quadratic formula with a square root is the most direct path. If Delta is negative, the equation has no real solutions and you need complex arithmetic. In engineering and physics problems this quick check is especially useful: a negative discriminant in a projectile-height equation means the projectile never reaches that height; a zero discriminant means it just barely reaches it.
Discriminant sign, root count and root type (quadratic)
| Discriminant | Real roots | Root type | Parabola |
|---|---|---|---|
| Delta > 0, perfect square | 2 | Two distinct rational | Crosses x-axis twice |
| Delta > 0, not perfect square | 2 | Two distinct irrational | Crosses x-axis twice |
| Delta = 0 | 1 (repeated) | One rational (repeated) | Tangent to x-axis |
| Delta < 0 | 0 | Two complex conjugates | Does not cross x-axis |
How the value of Delta = b^2 - 4ac determines the solutions of ax^2 + bx + c = 0.
Frequently asked questions
What is the discriminant of a quadratic equation?
The discriminant is the quantity Delta = b^2 - 4ac found inside the square root of the quadratic formula. It classifies the roots without requiring you to solve the full equation. A positive value means two real roots, zero means one repeated real root, and a negative value means two complex conjugate roots.
What does a negative discriminant mean?
A negative discriminant means the quadratic has no real roots. The square root of a negative number is imaginary, so the two solutions are complex conjugates of the form p + qi and p - qi. The parabola sits entirely above or below the x-axis and never crosses it.
Can I find the actual roots from the discriminant alone?
Not the roots themselves, but the discriminant tells you how many and what kind. To get the numerical root values you also need b and a: x = (-b +/- sqrt(Delta)) / 2a. This calculator computes all of that for you when you enable the "Compute root values" toggle.
What makes roots rational versus irrational?
When all coefficients a, b, c are integers and the discriminant is a perfect square (0, 1, 4, 9, 16, 25, ...), the square root in the quadratic formula is a whole number and both roots reduce to exact fractions. If the discriminant is a non-square positive integer the roots contain an irrational square root and cannot be written as exact fractions.
How does the cubic discriminant work?
For ax^3 + bx^2 + cx + d = 0 the discriminant is Delta = 18abcd - 4b^3d + b^2c^2 - 4ac^3 - 27a^2d^2. Positive: three distinct real roots. Zero: at least two roots coincide (repeated root). Negative: one real root and two complex conjugates. Unlike the quadratic case, the discriminant alone does not give you the roots - you need Cardano's formula or a numerical method.
What is the axis of symmetry and how is it related to the discriminant?
The axis of symmetry is the vertical line x = -b / 2a that divides the parabola into two mirror halves. It is also the x-coordinate of the vertex. When the discriminant is positive, the two roots are located symmetrically on either side of this axis. When the discriminant is zero the single repeated root equals the axis of symmetry exactly.
What is the vertex of a quadratic and how do I find it?
The vertex is the highest or lowest point of the parabola. Its x-coordinate is h = -b / 2a (the axis of symmetry) and its y-coordinate is k = a*h^2 + b*h + c. If a > 0 the vertex is a minimum; if a < 0 it is a maximum. The vertex y-value k equals -(Delta) / 4a, so the discriminant and the vertex are directly linked.