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01 // Chart Studio02 // Visual Solvers03 // Calculators04 // Formulas05 // Unicode & Symbols
03 // Bidirectional STEM Calculators // MATHΔ = b² - 4 × a × c

Quadratic Discriminant & Root Nature Solver

The quadratic discriminant Δ = b² - 4ac under the radical of the quadratic formula determines whether the polynomial ax² + bx + c = 0 has two distinct real roots (Δ > 0), one repeated real root (Δ = 0), or complex conjugate roots (Δ < 0).

Δ = b² - 4 × a × c
Pick which value you want to calculate (or type into any field):Solving for: Δ — Quadratic Discriminant
Calculated

Value of b² - 4ac

-5,000.005,000.00
unit

Leading coefficient of x² (a ≠ 0)

-100100
unit

Middle coefficient of x

-200200
unit

Y-intercept constant term

-200200
unit
Calculated Result · Quadratic Discriminant (Δ)
25

Step-by-Step Calculation

01Canonical Law: Δ = b² - 4 × a × c
02Given Inputs (Base Units): a = 2, b = 7, c = 3
03Solve for Δ (Quadratic Discriminant): Δ = 25

The quadratic discriminant Δ = b² - 4ac under the radical of the quadratic formula determines whether the polynomial ax² + bx + c = 0 has two distinct real roots (Δ > 0), one repeated real root (Δ = 0), or complex conjugate roots (Δ < 0).

Real-World Reference Values

Δ > 0 (Positive Discriminant)2 Distinct Real Roots

Parabola crosses the x-axis at (-b ± √Δ) / (2a)

Δ = 0 (Zero Discriminant)1 Repeated Real Root (Tangent)

Vertex sits directly on the x-axis at x = -b / (2a)

Δ < 0 (Negative Discriminant)2 Complex Conjugate Roots

Parabola never intersects the real x-axis

20-Row Comparison Table

Quadratic Discriminant (Δ) · Quadratic Coefficient (x²) (a)

Quadratic Discriminant & Root Nature Solver — Quadratic Coefficient (x²) → Quadratic Discriminant
#Quadratic Coefficient (x²) (a)Quadratic Discriminant (Δ)Δ %
01-20.5 295 +1080.00%
02-17.8684 263.4211 +953.68%
03-15.2368 231.8421 +827.37%
04-12.6053 200.2632 +701.05%
05-9.9737 168.6842 +574.74%
06-7.3421 137.1053 +448.42%
07-4.7105 105.5263 +322.11%
08-2.0789 73.9474 +195.79%
090.552632 42.3684 +69.47%
102 Current25 0.00%
115.8158 -20.7895 -183.16%
128.4474 -52.3684 -309.47%
1311.0789 -83.9474 -435.79%
1413.7105 -115.5263 -562.11%
1516.3421 -147.1053 -688.42%
1618.9737 -178.6842 -814.74%
1721.6053 -210.2632 -941.05%
1824.2368 -241.8421 -1067.37%
1926.8684 -273.4211 -1193.68%
2029.5 -305 -1320.00%
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Frequently Asked Questions

How do I find the constant c that makes a quadratic a perfect square trinomial?+

Click SOLVE FOR on c and set Δ = 0. The engine computes c = b² / (4a), completing the square.

What are the actual roots x₁ and x₂ once Δ is known?+

The roots are x₁ = (-b + √Δ) / (2a) and x₂ = (-b - √Δ) / (2a). For the default (2x² - 7x + 3 = 0, Δ = 25), the roots are x = 3 and x = 0.5.

What if Δ is a perfect square and a, b, c are integers?+

When Δ is a perfect square (such as 25), the quadratic factors cleanly over the rational numbers.