Quadratic Equation Solver | Free ax² + bx + c = 0 Tool
Solve any quadratic equation ax² + bx + c = 0. Find the discriminant, real or complex roots, vertex coordinates, and see the complete step-by-step solution.
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Understanding Quadratic Equations
The Discriminant (Δ = b² - 4ac)
- • Δ > 0: Two distinct real roots — the parabola crosses the x-axis twice
- • Δ = 0: One repeated root — the parabola touches the x-axis once
- • Δ < 0: Two complex roots — the parabola never crosses the x-axis
The Quadratic Formula
This formula works for any quadratic equation. The ± gives two solutions: one with + and one with -.
Reviewed by Emily Watson, M.Ed.
Mathematics EducatorMathematics teacher and curriculum designer with 15+ years of experience.
Quadratic Equation Solver Pros & Cons
Pros
- ✅ Real & complex root support
- ✅ Discriminant & vertex display
- ✅ Step-by-step solution shown
- ✅ Free forever — no subscriptions
- ✅ Works offline after page load
Cons
- ✗ Quadratic equations only (ax² + bx + c)
- ✗ No graphical parabola plot
Frequently Asked Questions
What is the quadratic formula?
x = (-b ± √(b² - 4ac)) / 2a. It solves any equation of the form ax² + bx + c = 0.
How do you know how many solutions a quadratic has?
Calculate the discriminant (b² - 4ac). Positive = 2 real roots, zero = 1 repeated root, negative = 2 complex roots.
What is the vertex of a parabola?
The vertex is the turning point. x = -b/(2a), y = f(x). It's the maximum if a < 0, minimum if a > 0.
Can a quadratic equation have no real solutions?
Yes, when the discriminant is negative. The solutions are complex numbers with imaginary parts.
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