Calculate the discriminant b²−4ac of a quadratic equation, determine the number and type of roots, find vertex and axis of symmetry, and see real or complex solutions.
The discriminant is the expression b² − 4ac that appears under the square root in the quadratic formula. It is one of the most powerful diagnostic tools in algebra because its sign alone tells you the nature of a quadratic equation's solutions without solving the equation. A positive discriminant means the parabola crosses the x-axis at two distinct points, giving two real roots. A discriminant of zero means the parabola is tangent to the x-axis, giving exactly one repeated root. A negative discriminant means the parabola never touches the x-axis, and the solutions are complex conjugates.
Beyond root classification, the discriminant connects to the geometry of the parabola. The vertex lies at x = −b/(2a) and represents the minimum or maximum of the function, while the axis of symmetry is the vertical line through the vertex. Vieta's formulas provide an algebraic shortcut: the sum of the roots equals −b/a and the product equals c/a, regardless of whether the roots are real or complex.
This calculator computes the discriminant from the coefficients a, b, and c, classifies the roots, solves for them (real or complex), and shows the vertex, axis of symmetry, y-intercept, and Vieta's formulas. The color-coded visualization makes it easy to see at a glance which root-type region you fall in, while the function-values table shows the parabola's shape across several x values.
Discriminant Calculator helps you solve discriminant problems quickly while keeping each step transparent. Instead of redoing long algebra by hand, you can enter Coefficient a, Coefficient b, Coefficient c once and immediately inspect Discriminant (Δ), Root Type, Root 1 (x₁) to validate your work.
This is useful for homework checks, classroom examples, and practical what-if analysis. You keep the conceptual understanding while reducing arithmetic mistakes in multi-step calculations.
Discriminant: Δ = b² − 4ac. Roots: x = (−b ± √Δ) / (2a). Vertex: (−b/(2a), f(−b/(2a))). Sum of roots: −b/a. Product of roots: c/a.
Result: Discriminant (Δ) shown by the calculator
Using the preset "x²+5x+6", the calculator evaluates the discriminant setup, applies the selected algebra rules, and reports Discriminant (Δ) with supporting checks so you can verify each transformation.
This calculator takes Coefficient a, Coefficient b, Coefficient c and applies the relevant discriminant relationships from your chosen method. It returns both final and intermediate values so you can audit the process instead of treating it as a black box.
Start with the primary output, then use Discriminant (Δ), Root Type, Root 1 (x₁), Root 2 (x₂) to confirm signs, magnitude, and internal consistency. If anything looks off, change one input and compare the updated outputs to isolate the issue quickly.
A strong workflow is manual solve first, calculator verify second. Repeating that loop improves speed and accuracy because you learn to spot common setup errors before they cost points on multi-step algebra problems.
The discriminant tells you how many real solutions a quadratic equation has. Positive means two real roots, zero means one repeated root, and negative means no real roots (two complex conjugate roots).
Yes. A negative discriminant (Δ < 0) means the quadratic has no real roots. The two solutions are complex numbers of the form a + bi and a − bi.
For a quadratic ax² + bx + c = 0, the discriminant is Δ = b² − 4ac. It is the expression under the square root in the quadratic formula.
If the discriminant is a perfect square and the coefficients are integers, the quadratic can be factored over the integers. If not, the roots are irrational or complex.
Vieta's formulas relate the roots of a polynomial to its coefficients. For a quadratic, the sum of roots is −b/a and the product is c/a.
If a = 0, the equation is not quadratic — it becomes bx + c = 0 (linear). The quadratic formula and discriminant apply only when a ≠ 0.