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Get NordVPN →Quadratic Equation Solver (ax² + bx + c = 0)
Enter coefficients a, b, c. The solver applies the quadratic formula x = (−b ± √(b²−4ac)) / 2a. Distinguishes two real roots, one repeated root, two complex conjugate roots, or degenerate cases.
1x² + -3x + 2= 0
Two real roots
x₁ = 2
x₂ = 1
- Discriminant (b² − 4ac)
- 1
- Parabola vertex
- (1.5, -0.25)
Related tools
How it works
The quadratic formula
For ax² + bx + c = 0 with a ≠ 0, the solutions are x = (−b ± √(b² − 4ac)) / (2a). The expression under the square root is the 'discriminant' (D = b² − 4ac), and its sign determines the type of roots.
D > 0: two distinct real roots. The parabola crosses the x-axis at two points.
D = 0: one repeated real root. The parabola touches the x-axis at the vertex (tangent).
D < 0: two complex conjugate roots. The parabola doesn't touch the x-axis at all.
Vertex and parabola shape
Every quadratic graphs as a parabola. The vertex (turning point) is at x = −b / 2a, and substituting back gives the y-coordinate. We compute it as y = −D / 4a, which is equivalent.
If a > 0 the parabola opens upward and the vertex is the minimum. If a < 0, downward and the vertex is the maximum. The axis of symmetry passes through the vertex vertically.
Degenerate cases
If a = 0, the equation isn't actually quadratic — it becomes linear: bx + c = 0, with solution x = −c/b (if b ≠ 0). We detect this and solve as linear.
If a = 0 AND b = 0: c must equal 0 for any solution to exist. If c = 0, every x is a solution; if c ≠ 0, no solution exists. We report both cases.
Frequently asked questions
›Why is the discriminant useful?
It tells you the nature of the roots without solving: D > 0 means two real roots, D = 0 means one repeated, D < 0 means complex. Often that's all you need to know.
›What's a 'repeated root'?
When D = 0, the formula gives x = −b/2a only. Algebraically, the equation factors as a(x − r)² = 0, so r appears twice as a root with 'multiplicity 2'.
›Are complex roots real-world useful?
Yes. AC circuits, signal processing, quantum mechanics, and aerodynamics all use complex numbers. Even when the physical answer is real, complex intermediate steps are common.
›Can I solve cubic or higher equations here?
Not in this tool. Cubics and quartics have closed-form solutions but they're more involved. For numerical solutions of high-degree polynomials, use NumPy or a CAS like Sage/Mathematica.
›What if my coefficients are very large?
Floating-point precision degrades for D when b² and 4ac are nearly equal. For research-grade accuracy, use a library with arbitrary-precision arithmetic.
›What does 'vertex' mean for a parabola?
The single 'turning point' where the parabola changes direction (from decreasing to increasing or vice versa). Located at x = −b/(2a). Useful for finding minima/maxima in optimization problems.
›Why are roots called 'roots'?
Historical: 'root' translates the Latin radix, used metaphorically as the source/origin of the equation. The roots are where the polynomial equals zero.
›Does the data leave my browser?
No. Calculation runs locally; nothing is sent to a server.
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