Quadratic Solver
Solve any quadratic a·x² + b·x + c = 0 for real or complex roots, with discriminant and vertex.
🔒 Runs entirely in your browser — nothing is uploadedHow the quadratic formula works
A quadratic equation has the standard form a·x² + b·x + c = 0, where a is not zero. The two solutions come from the quadratic formula x = (−b ± √(b² − 4ac)) / (2a). The quantity under the square root, b² − 4ac, is called the discriminant and it controls how many real solutions exist. This solver evaluates the formula directly and formats the result so you can read both roots at a glance, along with the discriminant and the vertex of the parabola.
When the discriminant is positive, the parabola crosses the x-axis at two distinct points, giving two real roots. When it is exactly zero, the parabola just touches the axis and the two roots merge into a single repeated root. When the discriminant is negative the square root of a negative number is imaginary, so the roots form a complex-conjugate pair written as p ± q·i. The solver detects each of these cases automatically.
Vertex and shape
Every quadratic graphs as a parabola whose turning point is the vertex. The vertex x-coordinate is −b ÷ (2a), and substituting it back into the equation gives the vertex y-coordinate. If a is positive the parabola opens upward and the vertex is the minimum; if a is negative it opens downward and the vertex is the maximum. Knowing the vertex helps you sketch the curve quickly and understand where the function reaches its extreme value, which is useful in optimization and physics problems such as projectile motion.
Accuracy and privacy
Calculations use standard double-precision arithmetic, which is accurate for typical coefficients used in homework, engineering estimates and finance. Extremely large or tiny coefficients can introduce small rounding differences, so round the displayed roots sensibly for your context. The entire computation happens locally in your browser. No coefficients are uploaded, nothing is stored and no external service is contacted, so you can solve as many equations as you like privately and instantly.
How to use
- Enter the coefficientsType a, b and c from the standard form a·x² + b·x + c = 0. a cannot be zero.
- Read the rootsSee both roots, real or complex, computed from the quadratic formula.
- Check discriminant and vertexThe discriminant tells you the root type; the vertex gives the parabola's turning point.