Solve any second-degree equation of the form ax² + bx + c = 0. Get complete step-by-step discriminant derivations, real or complex roots, vertex coordinates, and an interactive plotted parabola.
Must not be zero (a ≠ 0)
Coefficient "a" is required and cannot be zero — a quadratic equation needs a ≠ 0.
A quadratic equation is a second-order polynomial equation in a single variable x, where the highest exponent of the variable is two. The standard mathematical form is:
Here, a is the quadratic coefficient, b is the linear coefficient, and c is the constant term. When plotted on a coordinate Cartesian plane, every quadratic equation produces a smooth U-shaped curve known as a parabola.
The universal algebraic formula used to find the roots of any quadratic equation is derived by completing the square on the standard equation:
The expression underneath the square root, Δ = b² - 4ac, is called the discriminant. It reveals the nature of the solutions without having to complete the entire formula:
Identify coefficients: a = 1, b = -5, c = 6.
Identify coefficients: a = 1, b = 4, c = 13.
No. If a = 0, the x² term disappears, leaving bx + c = 0, which is a linear equation rather than a quadratic equation. The quadratic formula requires division by 2a, which would lead to an undefined division by zero if a = 0.
The vertex is the extreme turning point of the parabola. If a > 0, the parabola opens upward and the vertex represents the absolute minimum value. If a < 0, the parabola opens downward and the vertex represents the absolute maximum value. The x-coordinate of the vertex is always h = -b / (2a).
If a quadratic equation has real roots r₁ and r₂, it can be factored into a(x - r₁)(x - r₂) = 0. By the Zero Product Property, setting either factor to zero yields the solutions x = r₁ or x = r₂.
Factoring a quadratic uses common factors — find the GCF and LCM of any pair of integers instantly.
Trigonometric ratios share the discriminant-free side of algebra — solve sides, angles, area, and perimeter.
Quadratic roots are often fractions — add, subtract, simplify, and convert them step by step.
Explore the full college algebra toolkit — equations, functions, and factoring helpers.
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