Quadratic Equation Solver

Solve quadratic equations of the form ax² + bx + c = 0 with step-by-step solutions, discriminant analysis, and interactive parabola visualization.

Enter Coefficients

Your Equation
ax² + bx + c = 0
Cannot be 0
Example Equations
Parabola Graph
x y
Vertex
Roots
Y-intercept
Coefficient 'a' cannot be zero. For a linear equation (when a=0), use a linear equation solver.

Discriminant

b² - 4ac --
Nature of Roots --

Solutions

x1 --
x2 --

Parabola Properties

Vertex --
Axis of Symmetry --
Y-intercept --
Opens --

Root Properties

Sum of Roots --
Product of Roots --
Equation ax² + bx + c = 0
Solution(s) --

Step-by-Step Solution

Quadratic Formula Reference

Quadratic Formula

x = (-b +/- sqrt(b² - 4ac)) / 2a
Solves any quadratic equation ax² + bx + c = 0

Discriminant

D = b² - 4ac
D > 0: two real roots | D = 0: one root | D < 0: complex roots

Vertex Formula

x = -b/2a, y = f(-b/2a)
The vertex is the maximum or minimum point of the parabola

Sum of Roots

x1 + x2 = -b/a
Sum of the roots equals negative b over a

Product of Roots

x1 * x2 = c/a
Product of the roots equals c over a

Vertex Form

y = a(x - h)² + k
Where (h, k) is the vertex of the parabola

Frequently Asked Questions

What is the quadratic formula?

The quadratic formula is x = (-b +/- sqrt(b^2 - 4ac)) / 2a. It solves any quadratic equation of the form ax^2 + bx + c = 0, where a is not equal to 0. Enter your coefficients and this calculator will solve it instantly.

What is the discriminant and what does it tell us?

The discriminant is b^2 - 4ac. If it's positive, the equation has two distinct real roots. If it's zero, there's one repeated real root. If it's negative, the equation has two complex conjugate roots.

How do I find the vertex of a parabola?

The vertex of a parabola y = ax^2 + bx + c is at the point (-b/2a, f(-b/2a)). The x-coordinate is -b/2a, and the y-coordinate is found by substituting this x value back into the equation.

What are complex roots?

Complex roots occur when the discriminant is negative. They are written in the form a + bi and a - bi, where i is the imaginary unit (sqrt(-1)). Complex roots always come in conjugate pairs.