Quadratic Equation Solver
Solve any quadratic equation ax² + bx + c = 0. Get both roots — real or complex — and see the discriminant and vertex.
What is Quadratic Equation Solver?
A quadratic equation has the form ax² + bx + c = 0 (with a ≠ 0). This calculator uses the quadratic formula to find both roots — real or complex — in a single step, and also shows the discriminant and the vertex of the parabola y = ax² + bx + c.
Formula
The quadratic formula:
x = (−b ± √(b² − 4ac)) / 2a
The discriminant D = b² − 4ac tells you what kind of roots to expect:
- D > 0 → two distinct real roots
- D = 0 → one repeated real root
- D < 0 → two complex conjugate roots
The vertex (h, k) of the parabola is at h = −b/(2a), k = ah² + bh + c.
Worked example
Solve x² − 3x + 2 = 0: a=1, b=−3, c=2.
- D = 9 − 8 = 1 > 0 → two real roots
- x = (3 ± 1) / 2
- x₁ = 2, x₂ = 1
- Vertex: h = 1.5, k = −0.25 → (1.5, −0.25)
How to use this calculator
- Write your equation in standard form ax² + bx + c = 0. Move everything to the left of the equals sign if needed.
- Enter the coefficients. If a term is missing, set it to 0.
- The calculator returns both roots, the discriminant, the nature of the roots, and the vertex.
Frequently asked questions
What if a = 0?
Then the equation is not quadratic; it becomes bx + c = 0, a linear equation with one root x = −c/b (if b ≠ 0). This calculator returns no result in that case — use a linear solver instead.
Why do I get a complex number answer?
When the discriminant is negative, the parabola does not cross the x-axis, so there are no real roots. The roots exist only in the complex plane, written as a real part ± imaginary part × i.
What is the vertex useful for?
For y = ax² + bx + c, the vertex is the parabola's minimum (if a > 0) or maximum (if a < 0). It is used in optimization problems — maximum profit, maximum height of a projectile, etc.
Can this solve any polynomial?
No, only degree-2 polynomials. For cubics (degree 3) or higher, different methods are needed.