Discriminant
The discriminant of a quadratic equation ax² + bx + c = 0 is the expression b² − 4ac, denoted by the Greek letter Δ (delta) or D. The sign of the discriminant determines the nature and number of roots: positive means two distinct real roots, zero means one repeated real root, and negative means two complex (non-real) conjugate roots. The discriminant is the part under the square root sign in the quadratic formula, making it the key indicator of solvability over the real numbers.
Key Formula
Δ = b² − 4ac
LaTeX: \Delta = b^2 - 4ac
| Symbol | Meaning | Unit |
|---|---|---|
| Δ | Discriminant value | dimensionless |
| b | Coefficient of x in ax² + bx + c = 0 | dimensionless |
| a | Coefficient of x² | dimensionless |
| c | Constant term | dimensionless |
Worked Example
Problem
Without solving, determine the nature of the roots of 3x² − 2x + 5 = 0.
Solution
Step 1: Identify a = 3, b = −2, c = 5. Step 2: Calculate the discriminant. Δ = b² − 4ac Δ = (−2)² − 4(3)(5) Δ = 4 − 60 Δ = −56 Step 3: Interpret the result. Since Δ = −56 < 0, the equation has no real roots. It has two complex conjugate roots. Step 4: Find roots using quadratic formula. x = (2 ± √(−56)) / 6 = (2 ± 2i√14) / 6 = (1 ± i√14) / 3
Answer
Δ = −56 < 0: two complex conjugate roots — no real solutions exist
Discriminant Conditions and Root Types
| Condition | Discriminant Value | Root Type | Parabola Behaviour |
|---|---|---|---|
| Δ > 0 (perfect square) | e.g. 25 | Two distinct rational roots | Crosses x-axis at two rational points |
| Δ > 0 (not perfect square) | e.g. 3 | Two distinct irrational roots | Crosses x-axis at two irrational points |
| Δ = 0 | 0 | One repeated real root | Touches (is tangent to) x-axis |
| Δ < 0 | e.g. −7 | Two complex conjugate roots | Does not intersect x-axis |
Interactive Tools
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Related Terms
Quadratic Formula
The quadratic formula is an algebraic formula that gives the solutions (roots) of any quadratic equation ax² + bx + c = 0 directly in terms of its coefficients a, b, and c. It is derived by completing the square on the general quadratic and is the most reliable method for solving quadratics, working even when factoring over integers is impossible. The formula also reveals the nature of the roots through the discriminant b² − 4ac.
Quadratic Equation
A quadratic equation is a polynomial equation of degree 2, meaning the highest power of the variable is 2, written in standard form as ax² + bx + c = 0 where a ≠ 0. Its graph is a parabola, and it can have two, one, or no real solutions depending on the value of the discriminant (b² − 4ac). Quadratic equations model projectile motion, area problems, and many optimisation scenarios in physics and engineering.
Polynomial
A polynomial is an algebraic expression consisting of variables and coefficients combined using addition, subtraction, and multiplication, where variables have non-negative integer exponents. The general form is aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀, where the highest exponent n is called the degree. Polynomials are used extensively in calculus, numerical analysis, and computer science for approximating functions and solving complex problems.
From Latin "discriminare" meaning "to distinguish or separate", derived from "discrimen" (a distinction or dividing point). The term was introduced into algebra to describe the expression that discriminates (distinguishes) between the different types of roots. Its modern algebraic meaning was formalised by British mathematician James Joseph Sylvester in the 19th century.