Math & Stats
Quadratic Equation Solver
Solve ax² + bx + c = 0 instantly. Get the discriminant, real or complex roots, vertex, axis of symmetry, and full step-by-step solution.
Enter coefficients: ax² + bx + c = 0
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Quadratic Formula & Step-by-Step
x = (-b ± √(b² - 4ac)) / (2a)
Discriminant: Δ = b² - 4ac
Δ > 0 → two distinct real roots
Δ = 0 → one repeated root
Δ < 0 → two complex conjugate roots
Vertex: (-b/2a, f(-b/2a))
Axis of symmetry: x = -b/2aFAQ
Frequently asked questions about quadratic equations
What is the quadratic formula?
The quadratic formula is x = (-b ± √(b²-4ac)) / 2a. It solves any equation of the form ax² + bx + c = 0, giving both roots regardless of whether they are real or complex.
What does the discriminant tell us?
The discriminant (b²-4ac) determines the nature of roots: if positive, there are two distinct real roots; if zero, one repeated real root; if negative, two complex conjugate roots.
How do I find the vertex of a parabola?
The vertex is at x = -b/(2a), and the y-coordinate is found by substituting back: y = a(-b/2a)² + b(-b/2a) + c. The vertex is the minimum point if a > 0, maximum if a < 0.
What is the axis of symmetry?
The axis of symmetry is the vertical line x = -b/(2a) that passes through the vertex. The parabola is symmetric about this line, meaning the two roots are equidistant from it.
Can a quadratic equation have no real solutions?
Yes, when the discriminant is negative (b²-4ac < 0), the equation has no real solutions. Instead, it has two complex conjugate roots of the form p ± qi where i = √(-1).
What determines if a parabola opens up or down?
The sign of coefficient 'a' determines direction: if a > 0, the parabola opens upward (U-shape) with a minimum at the vertex; if a < 0, it opens downward (∩-shape) with a maximum.