Roots of A Polynomial of Degree 2 Calculator

This calculator helps you find the roots of a quadratic polynomial (degree 2) using the quadratic formula. Whether you're solving equations for academic purposes or practical applications, this tool provides accurate results and visualizations to help you understand the solutions.

Introduction

A quadratic polynomial is a second-degree polynomial equation of the form:

ax² + bx + c = 0

where a, b, and c are constants, and a ≠ 0. The roots of the polynomial are the values of x that satisfy the equation. These roots can be real or complex numbers, depending on the discriminant.

The quadratic formula is a reliable method for finding the roots of any quadratic equation. This calculator implements the formula to provide quick and accurate results.

The Quadratic Formula

The quadratic formula is derived from completing the square and is expressed as:

x = [-b ± √(b² - 4ac)] / (2a)

Where:

a is the coefficient of x²
b is the coefficient of x
c is the constant term

The discriminant (D) is the part under the square root in the formula:

D = b² - 4ac

The discriminant determines the nature of the roots:

If D > 0: Two distinct real roots
If D = 0: One real root (a repeated root)
If D < 0: Two complex conjugate roots

How to Use the Calculator

Enter the coefficients a, b, and c of your quadratic equation in the input fields.
Click the "Calculate" button to compute the roots.
View the results, including the discriminant and the roots.
Use the visualization to better understand the roots.
Click "Reset" to clear the inputs and results.

Note: The coefficient a must not be zero. If a is zero, the equation is no longer quadratic.

Interpreting the Results

After calculating the roots, you'll see:

The discriminant value
The roots of the equation
A visualization of the quadratic function

The visualization helps you understand the relationship between the coefficients and the roots. The parabola will open upwards if a is positive and downwards if a is negative.

Worked Examples

Example 1: Two Real Roots

Consider the equation: x² - 5x + 6 = 0

Using the quadratic formula:

x = [5 ± √(25 - 24)] / 2 x = [5 ± 1] / 2

The roots are x = 3 and x = 2.

Example 2: One Real Root

Consider the equation: x² - 6x + 9 = 0

Using the quadratic formula:

x = [6 ± √(36 - 36)] / 2 x = 6 / 2 = 3

The root is x = 3 (a repeated root).

Example 3: Complex Roots

Consider the equation: x² + 2x + 5 = 0

Using the quadratic formula:

x = [-2 ± √(4 - 20)] / 2 x = [-2 ± √(-16)] / 2 x = [-2 ± 4i] / 2

The roots are x = -1 + 2i and x = -1 - 2i.

Frequently Asked Questions

What is the quadratic formula?: The quadratic formula is a method for solving quadratic equations of the form ax² + bx + c = 0. It is expressed as x = [-b ± √(b² - 4ac)] / (2a).
How do I know if a quadratic equation has real roots?: A quadratic equation has real roots if the discriminant (b² - 4ac) is greater than or equal to zero. If the discriminant is negative, the roots are complex.
Can I use this calculator for higher-degree polynomials?: No, this calculator is specifically designed for quadratic polynomials (degree 2). For higher-degree polynomials, you would need a different method or calculator.
What does the visualization show?: The visualization shows the graph of the quadratic function y = ax² + bx + c. The roots are the points where the graph intersects the x-axis.
Is the quadratic formula always accurate?: Yes, the quadratic formula is mathematically proven to be accurate for all quadratic equations where a ≠ 0.