Second-Degree Polynomial Calculator

Explore the properties of your second-degree polynomials with precision and ease.

Second-Degree Polynomial Calculator

This calculator is a tool specially designed for the detailed analysis of second-degree polynomials, quadratic functions essential for understanding parabolic curves. Their general form is `f(x) = ax^2 + bx + c`, where `a`, `b`, and `c` are real constants with `a ≠ 0`. These functions are fundamental in mathematics and have numerous applications in fields such as physics and economics. The calculator uses three input fields for the coefficients `a`, `b`, and `c`, allowing a complete and in-depth study of the corresponding polynomial.

Usage Guide

To use this calculator, simply enter the values of the coefficients `a`, `b`, and `c` of the polynomial in the dedicated fields. For example, to analyze the polynomial `f(x) = 2x^2 - 3x + 1`, enter 2, -3, and 1 respectively in the corresponding fields. The calculator will then provide a complete analysis, including the domain of definition, roots, points of intersection with the axes, direction of variation, image set, convexity/concavity, and the different forms of the polynomial.

Here are some guidelines for entering the polynomial coefficients:

  1. Accepted inputs are,
    • integers, example: 5, -7
    • fractions, example: 1/3 or -2/9
    • decimal values, example: 3.9 or -9.65
    • constants, example: pi or e
    • common functions, for example: sin(pi/5)
    • square root operator, example : input sqrt(3) or 3^0.5 for `sqrt(3)`
  2. To enter a product of two factors, use the * operator. For example: enter 2*pi and not 2pi.

Domain of Definition

The domain of definition of a second-degree polynomial is the set of all real numbers, denoted `]-∞, +∞[`. This means that the function is defined and takes real values for every real `x`.

Resolution of the Equation f(x) = 0

To find the roots of the polynomial, we solve the equation `f(x) = 0`. The discriminant `Δ = b^2 - 4ac` determines the number of solutions. If `Δ > 0`, there are two distinct real solutions; if `Δ = 0`, there is a double real solution; and if `Δ < 0`, there are no real solutions. For example, for `f(x) = x^2 - 5x + 6`, `Δ = (-5)^2 - 4*1*6 = 1`, so there are two real solutions.

Intersection with Axes

The points of intersection with the x-axis are the roots of the equation.

For the y-axis, the polynomial intersects this axis at `f(0) = c`.

For example, for `f(x) = x^2 - 5x + 6`, the intersections with the x-axis are at the points `(2,0)` and `(3,0)`, and with the y-axis at `(0,6)`.

Direction of Variation and Extremum

A second-degree polynomial always has an extremum: a minimum if `a > 0` and a maximum if `a < 0`. The vertex of the parabola is located at `x = -b/(2a)`. For example, for `f(x) = x^2 - 5x + 6`, the minimum is at `x = 5/2`.

Image Set

The image set indicates the values that `f(x)` can take. For a second-degree polynomial with `a > 0`, the image set is `[f(-b/(2a)), +∞[`, and with `a < 0`, it's `]-∞, f(-b/(2a))]`.

Convexity and Concavity

Convexity (U-shaped) occurs when `a > 0`, and concavity (inverted shape) when `a < 0`. This affects how the function behaves relative to its axis of symmetry.

Forms of the Polynomial

The factored form is useful for finding the roots, the canonical form for identifying the vertex, and the general form for an overview. For example, for `f(x) = x^2 - 5x + 6`, the factored form is `(x - 2)(x - 3)`, and the canonical form is `(x - 5/2)^2 - 1/4`.

Derivative and Integral

The derivative of a second-degree polynomial `f(x) = ax^2 + bx + c` is `f'(x) = 2ax + b`. It indicates the direction of variation of the function.

An integral of f(x) is `F(x) = (1/3)ax^3 + (1/2)bx^2 + cx`. It can be used to calculate the area under the curve of the polynomial.

Examples and Applications

For example, the polynomial `f(x) = 2x^2 - 3x + 1` models a parabolic trajectory in physics. In economics, a polynomial like `f(x) = -x^2 + 4x - 3` might represent profit as a function of production volume.

In-Depth Exploration

If you wish to obtain more geometric properties of the curve of the second-degree polynomial, which is a conic section (specifically a parabola), use the conics calculators mentioned in the 'See Also' section (at the bottom of the page), particularly this conic section calculator.

If you wish to plot the curve of the second-degree polynomial, use this graph plotter.


This calculator offers a window into the fascinating world of second-degree polynomials. By exploring these concepts, we gain a deeper understanding of mathematical applications in various fields.

See also

Quadratic Equation Solver
Function Calculators
Conic sections Calculator
Parabola Calculator
Equation and Inequalities Calculators
Mathematics Calculators