Enter coefficients for ax² + bx + c = 0
Discriminant
Step-by-Step Calculation
- 1Identify coefficients: a = 1, b = 5, c = 6
- 2Write the formula: D = b² − 4ac
- 3Substitute: D = (5)² − 4(1)(6)
- 4Calculate b²: (5)² = 25
- 5Calculate 4ac: 4 × 1 × 6 = 24
- 6Subtract: D = 25 − (24) = 1
Roots (Quadratic Formula)
x = (−b ± √D) / (2a)
x = (−(5) ± √1) / (2 × 1)
Interpretation
- D > 0— two distinct real roots ✓ (your result: 1)
- D = 0— one repeated real root (the parabola just touches the x-axis)
- D < 0— no real roots (two complex conjugate roots)
The parabola opens upward (a > 0) and crosses the x-axis at two points.
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Discriminant Formula Calculator
Enter a, b, c for ax² + bx + c = 0 — instantly get the discriminant (b² − 4ac), nature of roots, step-by-step working, and the roots themselves.
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Check your discriminant calculations for algebra homework, verify the nature of roots before solving, and follow the step-by-step working to understand the method.
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Quickly evaluate discriminants for quadratic equations that appear in calculus, linear algebra, and engineering mathematics problems.
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Practice identifying the nature of roots across many equations efficiently — enter coefficients, check your answer against the calculator, and move on.
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Evaluate discriminants that arise in characteristic equations, signal processing, and systems analysis without manual calculation.
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Learn the discriminant formula by experimenting with different values of a, b, and c and observing how the discriminant and root nature change in real time.
How It Works
Enter the three coefficients of your quadratic equation in the form ax² + bx + c = 0. The calculator instantly computes the discriminant using the formula D = b² − 4ac, shows a full step-by-step breakdown of the calculation, identifies the nature of the roots (two distinct real roots, one repeated root, or two complex roots), and then solves for the roots using the quadratic formula. Exact rational roots are shown as fractions; irrational roots are shown as decimal approximations to 6 decimal places.
Features
- Instant real-time calculation — updates as you type
- Full step-by-step discriminant working (6 numbered steps)
- Nature of roots with color-coded result (real, repeated, or complex)
- Exact rational roots shown as simplified fractions when possible
- Decimal approximations for irrational roots (6 d.p.)
- Complex root display in a ± bi form
- Quadratic formula shown with substituted values
- Parabola interpretation (opens up/down, x-axis crossings)
- Copy individual results or copy all results at once
- Handles negative coefficients, fractions, and decimals
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What Is the Discriminant?
The discriminant is the expression b² − 4ac that appears under the square root sign in the quadratic formula. For a quadratic equation in the standard form ax² + bx + c = 0, the discriminant tells you how many real solutions the equation has — and what type they are — before you go through the full effort of solving it.
The name comes from the Latin discriminare, meaning to distinguish or separate, because the discriminant distinguishes between the three possible cases for the roots of a quadratic equation. It was named and studied by mathematicians including Arthur Cayley and James Sylvester in the 19th century, though the underlying relationship had been known since at least the work of Al-Khwarizmi in the 9th century.
The Discriminant Formula
For any quadratic equation ax² + bx + c = 0 (where a ≠ 0):
D = b² − 4ac
This formula comes directly from the quadratic formula. When you write out x = (−b ± √(b² − 4ac)) / (2a), the expression under the square root is the discriminant. The square root of the discriminant must be a real number for the equation to have real roots, which is only possible when D ≥ 0.
The Three Cases: What the Discriminant Tells You
Case 1: D > 0 — Two Distinct Real Roots
When the discriminant is positive, the equation has two different real number solutions. The square root of D is a positive real number, so adding and subtracting it from −b gives two different values for x.
If D is also a perfect square (4, 9, 16, 25, …), then √D is rational, and both roots are rational numbers — they can be expressed as exact fractions. If D is not a perfect square, the roots are irrational (like 2 + √3) and cannot be expressed as exact fractions.
Case 2: D = 0 — One Repeated Real Root
When the discriminant is exactly zero, both roots are equal — the equation has one solution repeated twice. This happens because √0 = 0, so both ±0 give the same result: x = −b / (2a).
Geometrically, this means the parabola y = ax² + bx + c is tangent to the x-axis — it just touches it at exactly one point without crossing. This unique point is called a "double root" or "repeated root."
Case 3: D < 0 — Two Complex (Non-Real) Roots
When the discriminant is negative, the equation has no real solutions. You cannot take the square root of a negative number in the real number system, so both roots are complex numbers of the form a ± bi (where i = √−1).
Complex roots always come in conjugate pairs: if one root is p + qi, the other is always p − qi. Geometrically, this means the parabola y = ax² + bx + c does not intersect the x-axis at all — it either lies entirely above it (when a > 0) or entirely below it (when a < 0).
How to Calculate the Discriminant: Step-by-Step
Given the quadratic equation 3x² + 5x − 2 = 0, here is the full working:
- Identify coefficients: a = 3, b = 5, c = −2
- Write the formula: D = b² − 4ac
- Substitute: D = (5)² − 4(3)(−2)
- Calculate b²: (5)² = 25
- Calculate 4ac: 4 × 3 × (−2) = −24
- Subtract: D = 25 − (−24) = 25 + 24 = 49
D = 49 > 0, so there are two distinct real roots. Since 49 is a perfect square (7² = 49), both roots are rational: x = (−5 ± 7) / 6, giving x₁ = 2/6 = 1/3 and x₂ = −12/6 = −2.
The Discriminant and the Quadratic Formula
The full quadratic formula is:
x = (−b ± √(b² − 4ac)) / (2a)
The discriminant D = b² − 4ac is the expression under the radical sign. Once you know D, finding the roots is straightforward:
- If D > 0: x = (−b ± √D) / (2a) — two different values
- If D = 0: x = −b / (2a) — one value (since ±0 = 0)
- If D < 0: x = (−b ± i√|D|) / (2a) — two complex values
Calculating the discriminant first is useful because it tells you what type of answer to expect before you finish the full quadratic formula, saving you from discovering at the end that you need to take the square root of a negative number.
Discriminant in the Context of Parabolas
The quadratic equation ax² + bx + c = 0 can be interpreted as asking where the parabola y = ax² + bx + c crosses the x-axis (where y = 0). The discriminant tells you exactly what the parabola looks like relative to the x-axis:
- D > 0: The parabola crosses the x-axis at two distinct points. The roots are the x-coordinates of those intersection points.
- D = 0: The parabola is tangent to the x-axis — it touches but does not cross. The vertex of the parabola sits exactly on the x-axis.
- D < 0:The parabola does not touch the x-axis at all. If a > 0, it floats entirely above; if a < 0, it lies entirely below.
This geometric interpretation is why the discriminant is so powerful: it summarizes the relationship between a parabola and the x-axis in a single number.
Frequently Asked Questions
What does it mean when the discriminant is negative?
A negative discriminant means the quadratic equation has no real solutions. The roots exist in the complex number system as a pair of complex conjugates (a + bi and a − bi), but there are no real numbers that satisfy the equation. On a graph, the parabola does not cross the x-axis.
Can the discriminant be zero?
Yes. When D = 0, the equation has exactly one real solution, called a double or repeated root. The root is x = −b / (2a). This happens when the parabola is tangent to the x-axis — it touches it at exactly one point (the vertex).
What is a perfect square discriminant?
When D is a positive perfect square (1, 4, 9, 16, 25, …), √D is a whole number, so both roots are rational. This means you can express the exact roots as fractions. When D is positive but not a perfect square, the roots are irrational (they involve a square root that cannot be simplified to a fraction).
Does the discriminant work for equations with fractional or decimal coefficients?
Yes. The formula D = b² − 4ac works for any real values of a, b, and c. This calculator accepts decimals and negative values. The roots may be harder to express exactly, but the discriminant calculation is identical.
Is there a discriminant for higher-degree polynomials?
Yes — discriminants can be defined for polynomials of any degree, though the formulas become much more complex. For a cubic equation ax³ + bx² + cx + d = 0, the discriminant is D = 18abcd − 4b³d + b²c² − 4ac³ − 27a²d². For quadratics, the discriminant is the most commonly encountered version in education.
What happens when a = 0?
When a = 0, the equation is no longer quadratic — it becomes a linear equation bx + c = 0 (if b ≠ 0) or has no solution (if b = 0 and c ≠ 0) or infinitely many solutions (if b = 0 and c = 0). The discriminant formula does not apply; this calculator will warn you if you enter a = 0.