quadratic expression calculator

Enter values for a, b, and c in the expression ax² + bx + c. You can also provide an optional x value to evaluate the expression.

Current expression: f(x) = x²

Coefficient a (required)

Coefficient b (required)

Constant c (required)

Value of x (optional)

What Is a Quadratic Expression?

A quadratic expression is a second-degree polynomial written in the form ax² + bx + c, where a, b, and c are constants and a ≠ 0. Quadratic expressions appear throughout algebra, geometry, physics, economics, and engineering.

Graphically, every quadratic expression produces a parabola. If a > 0, the parabola opens upward. If a < 0, it opens downward.

What This Calculator Gives You

This tool does more than basic substitution. It calculates the key features of a quadratic expression so you can quickly analyze behavior and solutions.

Standard form of the expression
Discriminant value (b² - 4ac)
Real or complex roots
Vertex coordinates
Axis of symmetry
Direction the parabola opens
Optional function value f(x) for a chosen input

How to Use the Calculator

Step 1: Enter coefficients

Type values for a, b, and c. These define your quadratic expression.

Step 2: Optionally enter x

If you want to evaluate the expression at a specific point, provide an x value.

Step 3: Click Calculate

The result panel will display roots, vertex, and other properties immediately.

Understanding the Discriminant

The discriminant controls the root type:

D > 0: two distinct real roots
D = 0: one repeated real root
D < 0: two complex conjugate roots

Knowing this quickly tells you whether the parabola intersects the x-axis, touches it once, or never crosses it in the real plane.

Example

For a = 1, b = -3, and c = 2, the expression is:

f(x) = x² - 3x + 2

The discriminant is 1, so there are two real roots: x = 1 and x = 2. The vertex is at (1.5, -0.25), and the parabola opens upward.

Why Quadratic Tools Matter

Quadratics are foundational in STEM and business modeling. You can use this calculator to check homework, verify algebra steps, model projectile motion, find optimal values, or analyze area and cost functions.

Because the process is instant, it helps you focus on interpretation rather than arithmetic mistakes.

Quick Tips

Use decimal values if needed; the calculator supports them.
If a = 0, the expression is no longer quadratic, and the tool will treat it as linear where possible.
For best readability, results are rounded to practical precision.