Quadratic Equation Solver

Enter the coefficients a, b, and c for any quadratic equation ax² + bx + c = 0. The solver applies the quadratic formula, shows the discriminant analysis, and returns real or complex roots with complete step-by-step working. Also finds the vertex of the parabola.

___x² + ___x + ___ = 0

Enter coefficients above to solve the equation

How It Works

The quadratic formula solves any equation of the form ax² + bx + c = 0:

x = (−b ± √(b² − 4ac)) / (2a)

Step 1 — Calculate the discriminant Δ = b² − 4ac
The discriminant determines the nature of the roots:

Δ > 0: Two distinct real roots
x₁ = (−b + √Δ) / (2a)
x₂ = (−b − √Δ) / (2a)

Δ = 0: One repeated real root (touches x-axis at one point)
x = −b / (2a)

Δ < 0: Two complex (imaginary) roots
x = −b/(2a) ± i×√|Δ|/(2a)

Example: 2x² − 4x − 6 = 0
Δ = (−4)² − 4(2)(−6) = 16 + 48 = 64
√64 = 8
x₁ = (4 + 8) / 4 = 3
x₂ = (4 − 8) / 4 = −1

Parabola vertex: (−b/(2a), (4ac−b²)/(4a))
For the example: vertex = (1, −8)

Frequently Asked Questions

What is the discriminant?

The discriminant Δ = b² − 4ac is the expression under the square root in the quadratic formula. Its sign tells you how many real solutions exist: positive → two real roots, zero → one repeated root, negative → two complex roots (no real solutions).

What are complex roots?

Complex roots involve the imaginary unit i, where i = √(−1). They appear in conjugate pairs: a + bi and a − bi. Complex roots mean the parabola does not cross the x-axis — it is entirely above or below it depending on the sign of a.

What is the vertex of the parabola?

The vertex is the highest or lowest point of the parabola y = ax² + bx + c. The x-coordinate is −b/(2a) and the y-coordinate is c − b²/(4a). If a > 0, the vertex is the minimum; if a < 0, it is the maximum.

Can a be zero?

No. If a = 0, the equation becomes linear (bx + c = 0), not quadratic. The calculator requires a ≠ 0. For a linear equation, the solution is simply x = −c/b.

What is a repeated root?

When Δ = 0, the quadratic has exactly one real solution (counted twice). Graphically, the parabola is tangent to the x-axis — it just touches it at one point without crossing.